Counting quadrant walks via Tutte's invariant method

In the 1970s, William Tutte developed a clever algebraic approach, based on certain"invariants", to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks confined to the first quadrant is governed by similar equations, and has led in the past 15 years to a rich collection of attractive results dealing with the nature (algebraic, D-finite or not) of the associated generating function, depending on the set of allowed steps, taken in $\{-1, 0,1\}^2$. We first adapt Tutte's approach to prove (or reprove) the algebraicity of all quadrant models known or conjectured to be algebraic. This includes Gessel's famous model, and the first proof ever found for one model with weighted steps. To be applicable, the method requires the existence of two rational functions called invariant and decoupling function respectively. When they exist, algebraicity follows almost automatically. Then, we move to a complex analytic viewpoint that has already proved very powerful, leading in particular to integral expressions of the generating function in the non-D-finite cases, as well as to proofs of non-D-finiteness. We develop in this context a weaker notion of invariant. Now all quadrant models have invariants, and for those that have in addition a decoupling function, we obtain integral-free expressions of the generating function, and a proof that this series is D-algebraic (that is, satisfies polynomial differential equations).


Introduction
In the past 15 years, the enumeration of plane walks confined to the first quadrant of the plane (Figure 1) has received a lot of attention, and given rise to many interesting methods and results. Given a set of steps S ⊂ Z 2 and a starting point (usually (0, 0)), the main question is to determine the generating function Q(x, y; t) ≡ Q(x, y) = i,j,n 0 q(i, j; n)x i y j t n , where q(i, j; n) is the number of n-step quadrant walks from (0, 0) to (i, j), taking their steps in S. This is one instance of a more general question consisting in counting walks confined to a given cone. This is a natural and versatile problem, rich of many applications in algebraic combinatorics [BM11, CDD + 07, GZ92,Gra02,Kra89], queuing theory [AWZ93, CB83,FH84], and of course in enumerative combinatorics via encodings of numerous discrete objects (e.g. permutations, maps...) by lattice walks [Ber07,BGRR,GWW98,KMSW15,LSW17].
At the crossroads of several mathematical fields. Most of the recent progress on this topic deals with quadrant walks with small steps (that is, S ⊂ {−1, 0, 1} 2 ). Then there are 79 inherently different and relevant step sets (also called models) and a lot is known on the associated generating functions Q(x, y; t). One of the charms of these results is that their proofs involve an attractive variety of mathematical fields. Let us illustrate this by two results: • a certain group G of birational transformations associated with the model plays a crucial role in the nature of Q(x, y; t). Indeed, this series is D-finite (that is, satisfies three linear differential equations, one in x, one in y, one in t, with polynomial coefficients in x, y and t) if and only if G is finite. This happens for 23 of the 79 models. The Figure 1. Three models of walks in the quadrant. The series Q(x, y; t) is algebraic for the first one, but only D-algebraic for the second. For the third one, it is not even D-algebraic.
positive side of this result (D-finite cases) mostly involves algebra on formal power series [BM02,BMM10,Ges86,GZ92,Mis09]. The negative part relies on a detour via complex analysis and a Riemann-Hilbert-Carleman boundary value problem [Ras12,KR12], or, alternatively, on a combination of ingredients coming from probability theory and from the arithmetic properties of G-functions [BRS14]. The complex analytic approach also provides integral expressions of Q(x, y; t) in terms of Weierstrass' function and its reciprocal. • Among the 23 models with a D-finite generating function, exactly 4 are in fact algebraic over Q(x, y, t) (that is, Q(x, y; t) satisfies a polynomial equation with polynomial coefficients in x, y and t). For the most mysterious of them, called Gessel's model (Fig. 1, left), a simple conjecture appeared around 2000 for the numbers q(0, 0; n), but resisted many attempts during a decade. A first proof was then found, based on subtle (and heavy) computer algebra [KKZ09]. The algebraicity was only discovered a bit later, using even heavier computer algebra [BK10]. Since then, two other proofs have been given: one is based on complex analysis [BKR17], and the other is, at last, elementary [BM16a].
Classifying solutions of functional equations. Beyond the solution of a whole range of combinatorial problems, the enumeration of quadrant walks is motivated by an intrinsic interest in the class of functional equations that govern the series Q(x, y; t). These equations involve divided differences (or discrete derivatives) in two variables. For instance, for Kreweras' walks (steps , ←, ↓), there holds: Q(x, y) = 1 + txyQ(x, y) + t Q(x, y) − Q(0, y) x + t Q(x, y) − Q(x, 0) y . (1) This equation is almost self-explanatory, each term corresponding to one of the three allowed steps. For instance, the term t Q(x, y) − Q(0, y) x counts walks ending with a West step, which can never be added at the end of a walk ending on the y-axis. The variables x and y are sometimes called catalytic. Such equations (sometimes linear as above, sometimes polynomial) occur in many enumeration problems, because divided differences like x − 1 have a natural combinatorial interpretation for any generating function F (x). Examples can be found in the enumeration of lattice paths [BF02,BMM10,BMP00], maps [Bro65,BT64,CF16,Tut95], permutations [BM11,BM03,BGRR]... A complete bibliography would include hundreds of references.
Given a class of functional equations, a natural question is to decide if (and where) their solutions fit in a classical hierarchy of power series: rational ⊂ algebraic ⊂ D-finite ⊂ D-algebraic,  Gessel). Right: Four algebraic models with weights [KY15].
where we say that a series (say Q(x, y; t) in our case) is D-algebraic if it satisfies three polynomial differential equations (one in x, one in y, one in t). An historical example is Hölder's proof that the Gamma function is hypertranscendental (that is, not D-algebraic), based on the difference equation Γ(x + 1) = xΓ(x). Later, differential Galois theory was developed (by Picard, Vessiot, then Kolchin) to study algebraic relations between D-finite functions [vdPS03]. This theory was then adapted to q-equations, to difference equations [vdPS97], and also extended to D-algebraic functions [Mal04]. Let us also cite [DHRar] for recent results on the hypertranscendence of solutions of Mahler's equations.
Returning to equations with divided differences, it is known that those involving only one catalytic variable x (arising for instance when counting walks in half-plane) have algebraic solutions, and this result is effective [BMJ06]. Algebraicity also follows from a deep theorem in Artin's approximation theory [Pop86,Swa98]. Moreover, as described earlier in this introduction, the classification of quadrant equations (with two catalytic variables x and y) with respect to the first three steps of the hierarchy (2) is completely understood. One outcome of this paper deals with the final step: D-algebraicity.
Contents of the paper. We introduce two new objects related to quadrant equations, called invariants and decoupling functions. Both are rational functions in x, y and t. Not all models admit invariants or decoupling functions. We show that these objects play a key role in the classification of quadrant walks: • first, we prove that invariants exist if and only if the group of the model is finite (that is, if and only if Q(x, y; t) is D-finite; 23 cases). In this case, decoupling functions exist if and only if the so-called orbit sum vanishes (Section 4). This holds precisely for the 4 algebraic models (Figure 2, left). • In those 4 cases, we combine invariants and decoupling functions to give short and uniform proofs of algebraicity. This includes the shortest proof ever found for Gessel's famously difficult model, and extends to models with weighted steps [KY15], for which algebraicity was sometimes still conjectural (Sections 3 and 4). • Models with an infinite group have no invariant. But we define for them a certain (complex analytic) weak invariant, which is explicit. Then for the 9 infinite group models that admit decoupling functions, we give a new, integral free expression of Q(x, y; t) (Section 5). This expression implies that Q(x, y; t) is D-algebraic in x, y and t. We compute explicit differential equations in y for Q(0, y; t) (Section 6). Note that an extended abstract of this paper appeared in the proceedings of the FPSAC'16 conference (Formal power series and algebraic combinatorics [BBMR16]). Moreover, two recent preprints of Dreyfus et al. essentially complete the differential classification of quadrant walks by proving that the remaining 47 infinite group models are not D-algebraic in y (nothing is known regarding the length variable t) [DHRS17,DHRSon]. The proof relies on Galois theory for difference equations. The nature of Q(x, y), for quadrant models with small steps, can thus be summarized as follows: Decoupling function No decoupling function Rational invariant (⇔ Finite group) Algebraic D-finite transcendental No rational invariant (⇔ Infinite group) Not D-finite, but Not D-algebraic (in y) D-algebraic Note that the existence of invariants only depends on the step set S, but the existence of decoupling functions is also sensitive to the starting point: in Section 7, we describe for which points they actually exist. In particular, we show that some quadrant models that have no decoupling function (and are not D-algebraic) when starting at (0, 0) still admit decoupling functions when starting at other points. Even though we have not worked out the details, we believe them to be D-algebraic for these points. This paper is inspired by a series of papers published by Tutte between 1973 and1984 [Tut73,Tut84], and then surveyed in [Tut95], devoted to the following functional equation in two catalytic variables: G(x, y; t) ≡ G(x, y) = xq(q−1)t 2 + xy qt G(1, y)G(x, y)−x 2 yt G(x, y) − G(1, y) x − 1 +x G(x, y) − G(x, 0) y .
(3) This equation appears naturally when counting planar triangulations coloured in q colours. Tutte worked on it for a decade, and finally established that G(1, 0) is D-algebraic in t. One key step in his study was to prove that for certain (infinitely many) values of q, the series G(x, y) is algebraic, using a pair of (irrational) series that he called invariants [Tut95]. They are replaced in our approach by (rational) invariants and decoupling functions. After an extension of Tutte's approach to more general map problems [BBM11,BBM17], this is now the third time that his notion of invariants proves useful, and we believe it to have a strong potential in the study of equations with divided differences.

First steps to quadrant walks
We now introduce some basic tools in the study of quadrant walks with small steps (see e.g. [BMM10] or [Ras12]). A simple step-by-step construction of these walks gives the following functional equation: K(x, y)Q(x, y) = K(x, 0)Q(x, 0) + K(0, y)Q(0, y) − K(0, 0)Q(0, 0) − xy, where K(x, y) = xy t (i,j)∈S x i y j − 1 is the kernel of the model. It is a polynomial of degree 2 in x and y, which we often write as K(x, y) = a(y)x 2 + b(y)x + c(y) = a(x)y 2 + b(x)y + c(x).
Seen as a polynomial in y, the kernel has two roots Y 0 and Y 1 , which are Laurent series in t with coefficients in Q(x). If the series Q(x, Y i ) is well defined, setting y = Y i in (6) shows that If this holds for Y 0 and Y 1 , then This equation will be crucial in our paper. We define symmetrically the roots X 0 and X 1 of K(x, y) = 0 (when solved for x).
The group of the model, denoted by G(S), acts rationaly on pairs (u, v), which will typically be algebraic functions of the variables x, y and t. It is generated by the following two transformations: where the polynomials a, a, c and c are the coefficients of K defined by (5). Note that these transformations do not depend on t, although K does. Indeed, c(u) a(u) = (i,−1)∈S u i (i,1)∈S u i , and symmetrically for c(v)/ a(v). Both transformations are involutions, thus G(S) is a dihedral group, which, depending on the step set S, is finite or not. For instance, if S = {↑, ←, }, the basic transformations are and Ψ : (u, v) → (u, uv), withū := 1/u andv := 1/v, and they generate a group of order 6: wherex := 1/x andȳ := 1/y. One key property of the transformations Φ and Ψ is that they leave the step polynomial, namely where (a) n = a(a + 1) · · · (a + n − 1) is the rising factorial. This conjecture was proved in 2009 by Kauers, Koutschan and Zeilberger [KKZ09]. A year later, by a computer algebra tour de force, Bostan and Kauers [BK10] proved that the three-variate series Q(x, y; t) is not only D-finite, but even algebraic. Two other more "human" proofs have then been given [BKR17,BM16a]. Here, we give yet another proof based on Tutte's idea of invariants.
Lemma 1. Let us take x = t + t 2 (u +ū), where u is a new variable andū stands for 1/u. Then Y 0 and Y 1 are Laurent series in t with coefficients in Q(u), satisfying The series Y 0 and Y 1 simply differ by the transformation u →ū. For i ∈ {0, 1}, the series Q(x, Y i ) and Q(0, Y i ) are well defined as series in t (with coefficients in Q(u)).
Proof. The expansions of the Y i near t = 0 are found either by solving explicitly K(x, Y i ) = 0, or using Newton's polygon method [Abh90]. To prove the second point, let us write where q(a, b, c, d) is the number of quadrant walks consisting of a North-East steps, b East steps, c South-West steps and d West steps. Given that x and Y i are series in t with respective valuation α = 1 and γ = −1, the valuation of the summand associated with the 4-tuple For Q(x, Y i ) to be well defined, we want that for any n ∈ N, only finitely many 4-tuples (a, b, c, d) satisfy a + b c + d, a c and v(a, b, c, d) n. The above expression of v shows that a, b and c must be bounded (for instance by n), and the inequality a + b c + d bounds d as well. Hence Q(x, Y i ) is well defined. This implies that Q(0, Y i ) is also well defined, as Q(0, y) is just obtained by selecting in the expression of Q(x, y) the 4-tuples such that a + b = c + d.
Returning to the generalities of Section 2, we conclude from Lemma 1 that (8) holds: so that we can rewrite (8) as .
This should be compared to (11). In Tutte's terminology, the series L(y) is, as J(y), an invariant, but this time it is (most likely) irrational. The connection between J(y) and L(y) will stem from the following lemma, which states, roughly, that invariants with polynomial coefficients in y are trivial. for some n 0 0. For x = t + t 2 (u + 1/u), the series A(Y 0 ) and A(Y 1 ) are well defined Laurent series in t, with coefficients in Q(u). If they coincide, then A(y) is in fact independent of y.
Proof. By considering A(y) − A(0), we can assume that A(0) = 0. In this case, a(j, n)y j t n .
Assume that A(y) is not uniformly zero, and let (see Figure 4 for an illustration). The inequalities 1 j n/2+n 0 imply that m is finite, at least equal to 1 − 2n 0 . Moreover, only finitely many pairs (j, n) satisfy m = n − j and j n/2 + n 0 . Since j n/2 + n 0 , any series Y = c/t + O(1) can be substituted for y in A(y), and Applying this to Y 0 = u/t + O(1) and Y 1 =ū/t + O(1), and writing that A(Y 0 ) = A(Y 1 ), shows that P (u) := j 1 a(j, m + j)u j = j 1 a(j, m + j)ū j = P (ū).
Hence the polynomial P (u) must vanish, which is incompatible with the definition of m.
The series J and L defined by (11) and (13) do not satisfy the assumptions of the lemma, as their coefficients are rational in y with poles at y = 0, −1 (for J) and y = −1 (for L): with S(y) = t(1 + y)Q(0, y). Still, we can construct from them a series A(y) satisfying the assumptions of the lemma. First, we eliminate the simple pole of J at 0 by considering (L(y) − L(0))J(y), which still takes the same value at Y 0 and Y 1 . The coefficients of this series have a pole of order at most 3 at y = −1. By subtracting an appropriate series of the form aL(y) 3 + bL(y) 2 + cL(y), where a, b and c depend on t but not on y, we obtain a Laurent series in t satisfying the assumptions of the lemma: the polynomiality of the coefficients in y holds by construction, and the fact that in each monomial y j t n , the exponent of j is (roughly) at most half the exponent of n comes from the fact that this holds in S(y), due to the choice of the step set (a walk ending at (0, j) has at least 2j steps). Thus this series must be constant, equal for instance at its value at y = −1. In brief, (L(y) − L(0))J(y) = aL(y) 3 + bL(y) 2 + cL(y) + d for some series a, b, c, d in t. Expanding this identity near y = −1 determines the series a, b, c, d in terms of S, and gives the following equation.
Proposition 3. For J and L defined by (14), and S(y) = t(1 + y)Q(0, y), Replacing in (15) the series J and L by their expressions (in terms of t, y and Q(0, y)) gives for Q(0, y) a cubic equation, involving t, y, and three additional unknown series in t, namely A 1 := Q(0, 0), A 2 := Q(0, −1) and A 3 := Q y (0, −1). It is not hard to see that this equation defines a unique 4-tuple of power series, with Equations of the form Pol(Q(0, y), A 1 , . . . , A k , t, y) = 0 occur in the enumeration of many combinatorial objects (lattice paths, maps, permutations...). The variable y is often said to be a catalytic variable. Under certain hypotheses (which essentially say that they have a unique solution (Q, A 1 , . . . , A k ) in the world of power series, and generally hold for combinatorially founded equations), their solutions are always algebraic, and a procedure for solving them is given in [BMJ06].
Applying the procedure of [BMJ06] to the equation obtained above for Gessel's walks shows in particular that A 1 , A 2 and A 3 belong to Q(Z), where Z is the unique series in t with constant term 1 satisfying Z 2 = 1+256t 2 Z 6 /(Z 2 +3) 3 . Details on the solution are given in Appendix A.4. Let us mention that, in the other "elementary" solution of this model, one has to solve an analogous equation satisfied by R(x) [BM16a, Sec. 3.4]. Once Q(0, y) is proved to be algebraic, the algebraicity of Q(x, 0), and finally of Q(x, y), follow using (7) and (6).

Extensions and obstructions: uniform algebraicity proofs
We now formalize the three main ingredients in the above solution of Gessel's model: the rational invariant J(y) given by (11), the identity (12) expressing xY 0 − xY 1 as a difference G(Y 0 ) − G(Y 1 ), and finally the "invariant Lemma" (Lemma 2). We discuss the existence of rational invariants J, and of decoupling functions G, for all quadrant models with small steps. In particular, we relate the existence of rational invariants to the finiteness of the group G(S). Finally, we show that the above solution of Gessel's model extends, in a uniform fashion, to all quadrant models (possibly weighted) known or conjectured to have an algebraic generating function (see Figure 2). They are precisely those that have a rational invariant and a decoupling function. For one of them, we need an algebraic variant of the invariant lemma, which is described in Section 4.4.

Invariants
Definition 4. A quadrant model admits invariants if there exist rational functions I(x) ∈ Q(x, t) and J(y) ∈ Q(y, t), not both in Q(t), such that I(x) = J(y) as soon as the kernel K(x, y) vanishes. Equivalently, the numerator of I(x) − J(y), written as an irreducible fraction, contains a factor K(x, y). The functions I(x) and J(y) are said to be an x-invariant and a y-invariant for the model, respectively.
Our definition is more restrictive than that of Tutte [Tut95], who was simply requiring I(x) and J(y) to be series in t with rational coefficients in x (or y).
The existence of (rational) invariants is equivalent to the following (apparently weaker) condition, which is the one we met in Section 3 (see (11)).
Lemma 5. Assume that there exists a rational function J(y) ∈ Q(t, y) \ Q(t) such that J(Y 0 ) = J(Y 1 ) when Y 0 and Y 1 are the roots of the kernel K(x, y), solved for y. Then I(x) := J(Y 0 ) = J(Y 1 ) is a rational function of x, and (I(x), J(y)) forms a pair of invariants.
Proof. We have I(x) = (J(Y 0 ) + J(Y 1 ))/2, hence I(x) is a rational function of x and t as any symmetric function of the roots Y 0 and Y 1 . The property I(x) = J(Y i ) tells us that I(x) equals J(y) as soon as K(x, y) = 0. which is the condition of Definition 4.
Example. In Gessel's case, J(y) was given by (11), and we find We can also check that K(x, y) divides I(x) − J(y). Indeed, The factor K(x, y) shows that the pair (I(x), J(y)) forms a pair of invariants for the model {→, , ←, } obtained by reflection in a vertical line. 2 In fact, two models differing by a symmetry of the square have (or have not) invariants simultaneously. Since these symmetries are generated by the reflections in the main diagonal and in the vertical axis, it suffices to consider these two cases.
Lemma 6. Take a model S with kernel K(x, y) and its diagonal reflection S, with kernel K(x, y) = K(y, x). Then S admits invariants if and only if S does, and in this case a possible choice is I(x) = J(x) and J(y) = I(y). A similar statement holds for the vertical reflection S, with kernel K(x, y) = x 2 K(x, y), wherex := 1/x. A possible choice is then I(x) = I(x) and J(y) = J(y).
Proof. The proof is elementary.
We can now tell exactly which models admit invariants. Note that it is easy to decide whether a given pair (I, J) is a pair of invariants: it suffices to check whether I(x) − J(y) has a factor K(x, y). But the following result tells us how to construct such pairs. Theorem 7. A (possibly weighted) quadrant model S has rational invariants if and only if the associated group G(S) defined by (9) is finite.
Assume this is the case, and let H(x, y) be a rational function in Q(x, y, t). Consider the rational function H σ (x, y) := γ∈G(S) H(γ(x, y)). Then and J(y) = H σ (X 0 , y) are respectively rational functions in (t, x) and (t, y), and they form a pair of invariants, as long as they do not both belong to Q(t).
Proof. Assume that the model has invariants I(x), J(y), with I(x) ∈ Q(t). If (x , y ) is any element in the orbit of (x, Y 0 ), then K(x , y ) = 0, hence I(x ) = J(y ). But the form of Φ and Ψ implies by transitivity that I(x ) = I(x) and J(y ) = J(y).
Assume that the group of the model is infinite. Then the orbit of (x, Y 0 ) is infinite as well. Indeed, if it were finite, then there would exist γ ∈ G(S), different from the identity, such that γ(x, Y 0 ) = (x, Y 0 ). Denoting γ(x, y) = (r(x, y), s(x, y)), where both coordinates r and s are in Q(x, y), this would mean in particular that r(x, y) − x vanishes at y = Y 0 , forcing this rational function to zero, or to have a factor K(x, y) in its numerator. But this is impossible since r(x, y) does not involve the variable t (while K does), hence r(x, y) = x. By the same argument, s(x, y) = y, hence γ is the identity, which contradicts our assumption. Hence the orbit of (x, Y 0 ) is infinite. This implies that infinitely many series x occur in it (as the first coordinate of a pair), and thus the equation (in x ) I(x ) = I(x) has infinitely many solutions. This is clearly impossible since we have assumed that I(x) ∈ Q(t).
Conversely, take a model with finite group, a rational function H(x, y) in Q(x, y, t), and define H σ as above. For instance, for a model S with a vertical symmetry, G(S) has order 4, and the orbit of (x, Y 0 ) reads: Thus if we take H(x, y) = x, then H σ (x, y) = 2(x +x) and J(y) = 2(X 0 + X 1 ) = −2 b(y) a(y) . Returning to a general group, observe that H σ takes the same value, by construction, on all elements of the orbit of (x, y). In particular, H σ (x, Y 0 ) = H σ (x, Y 1 ). Hence the above defined series I(x) and J(y) are rational in x and t (or y and t). Moreover, J(Y 0 ), being the sum of H over the orbit of (x, Y 0 ), coincides with I(x), and (I, J) is a pair of invariants (unless I and J both depend on t only).
For instance, for the reverse Kreweras model {→, ↑, }, and H(x, y) = x, we find I(x) = J(y) = 1/t. But taking instead H(x, y) = 1/x gives true invariants: Let us finally prove that, for any weighted model, there exists k 1 such that the invariant I (k) (x) obtained from the function H (k) (x, y) = x k actually depends on x. Assume this is not the case. Let G(S) have order 2n, and let x 0 = x, x 1 , . . . , x n−1 be the n distinct series x that occur in the orbit of (x, Y 0 ) as the first coordinate of some pair. Then by assumption, for all k, which shows that all symmetric functions of the x i 's depend on t only. This implies that each x i is an algebraic function of t only, which is impossible since x 0 = x.
Since we mostly focus in this section on algebraic quadrant models, we have only given in Table 1 explicit invariants for the four algebraic (unweighted) models. The remaining 19 models with a finite group either have a vertical symmetry (in which case they admit I(x) = x +x as x-invariant), or differ from an algebraic model by a symmetry of the square (in which case Lemma 6 applies). Invariants for the four weighted models of Figure 2 are given in Table 3.
see Lemma 6 and the previous model Table 2. Rational invariants for weighted models.
In Section 5, we introduce a weaker notion of (possibly irrational) invariants, which guarantees that any non-singular quadrant model now has a weak invariant. One key difference with the algebraic setting of this section is that the new notion is analytic in nature.

Decoupling functions
We now return to the identity (12), which we first formalize into an apparently more demanding condition.
Definition 8. A quadrant model is decoupled if there exist rational functions F (x) ∈ Q(x, t) and G(y) ∈ Q(y, t) such that xy = F (x) + G(y) as soon as K(x, y) = 0. Equivalently, the numerator of F (x) + G(y) − xy, written as an irreducible fraction, contains a factor K(x, y). The functions F (x) and G(y) are said to form a decoupling pair for the model.
Again, this is equivalent to a statement involving a single function G(y), as used in the previous section (see (12)).
Lemma 9. Assume that there exists a rational function G(y) ∈ Q(y, t) such that where Y 0 and Y 1 are the roots of the kernel K(x, y), solved for y.
, and (F (x), G(y)) is a decoupling pair for the model.
Proof. We have hence F (x) is a rational function of x and t since it is symmetric in Y 0 and Y 1 . The property F (x) = xY i − G(Y i ) tells us precisely that the condition of Definition 8 holds.
Example. In Gessel's case, we had G(y) = −1/(t(1 + y)) (see (12)), corresponding to F (x) = Remark. By combining (8) and (16), we see that if both series Q(x, Y i ) are well defined, then with S(y) = K(0, y)Q(0, y). In Tutte's terminology, this would make S − G a second "invariant". But our terminology is more restrictive, as our invariants must be rational. Now, which of the 79 quadrant models are decoupled? Not all, at any rate: for any model that has a vertical symmetry, the series Y i are symmetric in x and 1/x, and so any expression of x of the form (G(Y 0 ) − G(Y 1 ))/(Y 0 − Y 1 ) would be at the same time an expression of 1/x.
In the case of a finite group, we give in Theorem 12 below a criterion for the existence of a decoupling pair, as well as an explicit pair when the criterion holds. This shows that exactly four of the 23 finite group models are decoupled (and these are, as one can expect from the algebraicity result of Section 3, those with an algebraic generating function). The four weighted models of Figure 2, right, are also decoupled.
For models with an infinite group, we have first resorted to an experimental approach to construct invariants. Indeed, one can look for decoupling functions by prescribing the form of the partial fraction expansion of G(y): we first set for fixed values of d, m, d 1 , . . . , d m . We then compute F (x) using (17) and write that the numerator of F (x) + G(y) − xy is divisible by K(x, y). This gives a system of equations relating the a i , α i,e and r i . Solving this system tells us whether the model has a decoupled pair for our choice of d, m and the d i .
In this way, we discovered nine decoupled models among the 56 that have an infinite group. We could then prove that there are no others. The following theorem summarizes our results.
Theorem 10. Among the 79 relevant quadrant models, exactly 13 are decoupled: the 4 models of Figure 2, left, and the 9 models of Table 4. Moreover, the 4 weighted models shown on the right of Figure 2 are also decoupled.
The rest of Section 4.2 is devoted to proving the above theorem. Tables 3 and 4 give explicit decoupling pairs, respectively for finite and infinite groups. One can easily check that they satisfy Definition 8. The key point is then to prove that there are no other (unweighted) decoupled models. To prove this, we consider separately the finite and infinite group cases.  In this case, we have found a systematic procedure to decide whether there exists a decoupling pair, and to construct one (when it exists). We consider in fact a more general problem, consisting in writing a rational function H(x, y) as F (x) + G(y) when the pair (x, y) cancels the kernel. The above definition of decoupled models deals with the case H(x, y) = xy, but the general case is not harder and allows us to consider starting points other than (0, 0). This is further discussed in Section 7. As we shall see, decoupling functions exist if and only if a certain rational function, called orbit sum, vanishes. Our approach adapts [FIM99, Thm. 4.2.9 and Thm. 4.2.10] to our context 1 Notation. We recall that the group G(S) is generated by the involutions Φ and Ψ defined by (9). Its elements consist of all alternating products of Φ and Ψ. We denote Θ = Ψ · Φ, and observe that Θ −1 = Φ · Ψ. Each element γ of the group has a sign, depending on the number of generators it involves: sign(Θ k ) = 1, while sign(Φ · Θ k ) = −1 for all k.
For A(x, y) ∈ Q(x, y), and ω = γ∈G(S) c γ · γ an element of the group algebra Q[G(S)], we denote This is again a rational function in x and y. A particular case is the notation H σ (x, y) already used in Theorem 7, corresponding to the case where σ is the sum of all elements of the group.
We now generalize Definition 8.
Definition 11. Given a quadrant model S, and its kernel K(x, y), a function H(x, y) ∈ Q(x, y) is decoupled if there exist F (x) ∈ Q(x, t) and G(y) ∈ Q(y, t) such that, for all pairs (x, y) canceling the kernel, H(x, y) = F (x) + G(y).
Theorem 12. Let S be a step set such that the associated group G(S) is finite of order 2n. Then H ∈ Q(x, y) is decoupled if and only if H α (x, y) = 0, where α is the following alternating sum: In this case, one can take The corresponding value of G is then G(y) = H τ (X 0 , y) + H τ (X 1 , y) + (1 − 1/(2n))J(y), where J(y) is the invariant defined in Theorem 7 and Proof. Assume that H is decoupled. Then for every pair (u, v) in the orbit of (x, Y 0 ), we have H(u, v) = F (u) + G(v) = H(γ(x, Y 0 )) for some element γ of G(S). Now recall that if (u , v ) = Φ(u, v), then v = v (and symmetrically for the transformation Ψ). Hence taking the alternating sum of H(u, v) = F (u) + G(v) over the orbit of (x, Y 0 ) shows that H α (x, Y 0 ) = 0, which implies that H α (x, y) is uniformly zero since Y 0 depends on t while x and H do not.
Suppose now that H α = 0, and define F (x) and G(y) as above. Note that Upon grouping the terms in Θ j , and those in Θ j Φ, we obtain where σ and α are respectively the sum, and the alternating sum, of the elements of G(S).
Returning to (18), and using H α = 0, we obtain that Proof of Theorem 10 in the finite group case. We now apply Theorem 12 to H(x, y) = xy and to the 23 models with a finite group (listed for instance in [BMM10, Sec. 8]). We find indeed that the alternating orbit sum of H(x, y) vanishes in four cases only. Applying to them the procedure of the theorem gives the decoupling functions of Table 3. The procedure applies as well to weighted models. We have sometimes replaced the pair (F, G) of Theorem 12 by the decoupling pair (F + cI, G − cJ), where (I, J) is a pair of invariants, to simplify the expression of G(y).

4.2.2.
The infinite group case. When G(S) is infinite, we have not found any criterion comparable to Theorem 12 that would decide whether the function H(x, y) = xy is decoupled. Our approach involves some case-by-case analysis, and relies on the following observations: , and xY i (x) becomes infinite at some x c (which may be finite or infinite, and depend on t), The idea is then to argue ad absurdum. We use the first observation to produce poles of F (or G), and the second to prove that F (or G) has infinitely many poles, which is uncommon for a rational function. Note that the idea of propagating poles is classical when searching for rational solutions of difference equations (see e.g. [Abr95]), and is also used in the quadrant context in [DHRS17].
Definition 13. Let K be the algebraic closure of C[[t]]. Its elements are Puiseux series in t with complex coefficients.
where u is the other root (if any) of the equation K(U, v) = 0 (solved for U ). We define symmetrically ψ(u, v). For X ∈ K, the equation K(X, y) has at most two solutions Y 0 (X) and Y 1 (X), which belong to K as well (we ignore infinite solutions). The x-orbit of X is the set of pairs in K 2 that can be obtained from (X, Y 0 (X)) or (X, Y 1 (X)) by repeated application of the transformations φ and ψ (as long as they are well defined).
We define the y-orbit of an element of K in a similar fashion.
The second example below shows that one may obtain more pairs in the x-orbit by using φ and ψ rather than Φ and Ψ.
Proof. It follows from the second observation above that each Y i (X) is a pole of G, and we propagate the property using the fact that φ and ψ preserve one coordinate.
We now apply these ideas to the proof of Theorem 10, in the infinite case. We distinguish several cases, depending on the value of a(x) = [x 2 ]K(x, y) = tx (i,1)∈S x i . The first two cases build on the observation that no decoupled model found so far has a(x) = t(1 + x 2 ) nor a(x) = t(1 + x + x 2 ). A Maple session, available on the authors' webpages, examines in details all non-decoupled models.
Method 1. When a(x) = t(1 + x 2 ). Let us consider for instance the three-step model { , , }. Then one of the series Y j (x) becomes infinite at x = i (the other takes the value it). This implies that either F is singular at i, or G is singular at ∞.
The x-orbit of X = i reads: Each x k , y k is rational in t, and it is easy to see, by induction on t, that x k has valuation 2k, while y k has valuation 2k + 1. Hence the x-orbit of i is infinite, and by Lemma 14, F cannot be singular at i. Moreover, the identity But the same argument applies with −i instead of i. This gives G(y) ∼ −iy as y → ∞, which contradicts the previous conclusion. Assuming that the model is decoupled has led us to a contradiction, hence it is not decoupled.
Since the above model is singular, let us now take a non-singular one, for instance { , , , ↓}. The above argument applies verbatim, except for the proof that the x-orbit of i is still infinite. Using again the notation (19), we have x 0 = i, y 0 = (1 − i)t, and the argument is now that This argument applies to all models such that a(x) = t(1 + x 2 ), and symmetrically, all those such that a(y) = t(1 + y 2 ). They are labeled 1a and 1b in Table 5, and we have thus proved non-decoupling for 17 models with an infinite group.
Method 2. When a(x) = t(1 + x + x 2 ). The key ingredient that we used above for a(x) = t(1 + x 2 ) is the fact that a(x) has two conjugate roots. A similar argument applies when these roots are j and 1/j, with j = e 2iπ/3 , and proves non-decoupling for 12 additional models, indicated in Table 5 by 2a and 2b (depending on whether the argument is applied to a(x) or a(y)).
Method 3. When a(x) = t(1 + x). Consider for instance the model S = {↑, , }, and assume that it has a decoupling pair (F, G). One of the roots Y i (x) is infinite at x = −1, thus either F is singular at −1, or G is singular at ∞. One can check that the x-orbit of −1 is infinite, thus only the latter property holds, and moreover G(y) ∼ −y as y → ∞. Now let x be real and tend to −∞.
This argument establishes non-decoupling for 12 more models, indicated in Table 5 with the labels 3a and 3b. Note that the concluding argument is slightly different for the three x/ysymmetric models among them, for instance {↑, , , →}. For these models, we can assume that F = G. The first part of the argument gives as before F (y) ∼ −y as y → ∞. In the second part, we have The argument is also slightly more involved for the (asymmetric) model S = {↑, , ←, , , →}. As before, the decoupling identity written around x = −1 and Y (x) → ∞ shows that G(y) ∼ −y as y → ∞. Then, writing it around x = −∞ and Y (x) ∼ −x shows that F (x) ∼ −x 2 as x → ∞, which yields no contradiction. But writing finally the decoupling identity around y = −1 and X(y) ∼ − 1+t t(1+y) implies that G is singular at −1, which is impossible because the y-orbit of −1 is infinite.
Method 4. When a(x) = tx. Consider now the model S = {↑, ←, ↓, }. We first note that the x-orbit of 0 is infinite, hence F cannot be singular at 0. Now, writing the decoupling identity around This argument establishes non-decoupling for 4 new models, labeled 4a in Table 5. Again, the argument is a bit different for the x/y-symmetric model {↑, ←, , ↓, →}. Here, we can assume that F = G. Expanding the decoupling identity around x = 0 and Y (x) ∼ −1/x shows that F is finite at ∞. But this is not compatible with the decoupling identity, written around 5. The last two models. We are left with the two symmetric "forks", S = {↑, , →, } and its reverse S = {←, , ↓, }, for which a(x) = tx(1 + x) and a(x) = tx 2 , respectively. In both cases, we assume F = G.
For the first model, expanding the decoupling identity around x = −1 and Y (x) → −∞ implies that F (y) ∼ −y as y → ∞ (we apply Lemma 14 at x = −1). Then, expanding the decoupling identity around For the second model, expanding the decoupling identity around x = 0 and Y (x) → −1/x 2 implies that F (y) ∼ −|y| 1/2 as y → −∞, which is impossible for a rational function.
We have now concluded the proof of Theorem 10.  1b  1b  3a  2a  2a  3a  1b  1a  3a  2a  2a  2b   2b  3a  1b  1b  1b  2a  1a  3a  2a  2a   Table 5. The 56 models with an infinite group. Exactly 9 are decoupled. For the others, we give a label that tells which method can be used to prove that it is not decoupled. These labels refer to the numbering in Section 4.2.2. We have put in the same cell models that only differ by a symmetry of the square.

The invariant lemma
At this stage, we have found 8 models (4 unweighted, 4 weighted), which, as Gessel's model, admit invariants and are decoupled. They are in fact the 8 algebraic (or conjecturally algebraic) models of Figure 2. In order to prove their algebraicity as we did in Section 3 for Gessel's model, we still need to adapt the third and last ingredient of Section 3, namely Lemma 2. We can do this for 7 of our 8 models. The resisting model is the reverse Kreweras model, with steps →, ↑, . We shall circumvent this difficulty in the next subsection.
Lemma 15 (The invariant lemma). Let S be one of the models of Figure 2, distinct from the reverse Kreweras model. If S is one of the last two models, set x = (1 + u)(1 +ū)t, with u = 1/u. Otherwise, set x = t + (u +ū)t β , where β is given in Table 6 below. Then around t = 0, the roots Y 0 and Y 1 of K(x, y) = 0 expand as and where γ is given by Table 6. The series Q(x, Y i ) and Q(0, Y i ) are well defined as series in t (or √ t when γ is a half-integer) with coefficients in Q(u).
Let A(y) be a Laurent series in t with polynomial coefficients in y, of the form where a(j, n) ∈ Q and ρ < 1/|γ| if γ < 0. Then A(Y 0 ) and A(Y 1 ) are well defined series in t (or √ t) with coefficients in Q(u). If they coincide, then A(y) is in fact independent of y.
The proof mimics the proofs of Lemmas 1 and 2 used in the Gessel case, where we had β = 2, γ = −1 and ρ = 1/2.  It now remains, for each of the 7 models for which the above lemma holds, to construct a series A(y) satisfying the conditions of the lemma, by combining the invariants of Tables 1 and 2  and the decoupling functions of Table 3. Applying Lemma 15 gives for each of these 7 models a polynomial equation of the form where the series A i are derivatives of Q(0, y) with respect to y, evaluated at y = 0 or y = −1. These equations are made explicit in Appendix A. The next step will be to solve them, using the general procedure of [BMJ06]. This is described in Section 4.5, and detailed in Appendix A, but we delay this description to establish an equation of the same form for the reverse Kreweras model, to which Lemma 15 does not apply.

An alternative to the invariant lemma
The above method fails for the reverse Kreweras model. The reason is that there exists no Puiseux series x in t, with coefficients in C(u), such that both Y 0 (x) and Y 1 (x) can be substituted for y in Q(x, y) (the proof is elementary, by considering the valuation of x in t). Hence we have no counterpart of Lemma 15. However, we will now show that this tool is not essential: if we replace the statements and by their more algebraic counterparts, according to which I(x) − J(y) and F (x) + G(y) − xy are both divisible by the kernel K(x, y), we obtain a substitution-free solution, which works as well for the reverse Kreweras walks.
To clarify this, let us first consider Kreweras' model, with steps , ←, ↓, for which the method that we have described in the previous subsections works. The kernel is and the functional equation reads with R(x) = txQ(x, 0) = S(x). The invariants and decoupling functions can be taken as and The fact that we can take I = J and F = G comes from the x/y-symmetry of the model. How does the method presented so far work? First, for x = t + (u +ū)t 5/2 , both Y 0 and Y 1 can be substituted for y in the functional equation, yielding Then, the decoupling function allows us to rewrite this as The invariant J satisfies the same equation as S − G: We now form a third series A satisfying this equation, but having no pole at y = 0 (nor t = 0): By the invariant lemma (Lemma 15), A(y) must be independent of y.
We now give a specialization-free version of this argument. What plays the role of (21) is simply the functional equation (20). The decoupling property stems from and allows us to rewrite the functional equation as This is the counterpart of (22). We multiply this equation by which gives: This should be compared to the invariant property, which stems from: We now derive from the last two equations an equation satisfied by the pole-free series A(y) defined by (23): Using the expressions of G(y) and S(y), we observe that C(x, y) is a formal power series in t with coefficients in Q[x, y]. This series, multiplied by the polynomial K(x, y), decouples as A(x)−A(y).
The following lemma shows that this is impossible, unless C(x, y) = 0 = A(x) − A(y). Thus we conclude that A(x) is independent of x, which was a consequence of the invariant lemma in our first approach.
Lemma 16. Consider a quadrant model and its kernel K(x, y). If there are series A(x), B(y) and and C(x, y) = 0.
Proof. We define a total order on monomials t n x i y j , for (n, i, j) ∈ N 3 , by taking the lexicographic order on (n, i, j). For a series S, we denote by min(S) the exponent (n, i, j) of the smallest monomial occurring in S. Then min K(x, y) = xy. Assume C(x, y) = 0, and let M be its minimal monomial. Then xyM is the minimal monomial of K(x, y)C(x, y), and should thus occur in A(x) − B(y), which is impossible.
We now adapt this to the reverse Kreweras model, with steps →, ↑, . The kernel is and the functional equation reads . The invariants and decoupling functions can be taken as and The decoupling property stems from and allows us to rewrite the functional equation as This should be compared to the invariant property We now cancel poles at y = 0 (and t = 0) by considering the series A linear combination of (24) and (25) gives (0)) .
Using the expressions of G(y) and S(y), we observe that C(x, y) is a series in t with coefficients in Q[x, y]. We conclude as in Kreweras' case that C(x, y) = 0 = A(x) − A(y), so that A(y) is independent of y. Expanding this series around y = 0 gives:

Effective solution of algebraic models
At this stage, for each of the eight models of Figure 2, we have obtained an equation of the form where the series A i are derivatives of Q(0, y) with respect to y, evaluated at y = 0 or y = −1.
Their exact forms are given in Appendix A. It remains to apply the general procedure of [BMJ06] to solve them. This is also detailed in the Appendix, and a Maple session supporting the calculations is available on the authors' webpages. These calculations are of course heavier when k is large: the most complicated models turn out to be Gessel's model and the last weighted model, for which k = 3 (we recall that this model was only conjectured to be algebraic [KY15]). For the reverse and double Kreweras models, and for the third weighted model, k = 2, while k = 1 for Kreweras' model and for the first two weighted models. In all cases the solution is (as already claimed) algebraic. In particular, the generating function Q(0, 0) of excursions has degree 3, 3, 4, 8, 6, 3, 3, 8 over Q(t), if we scan models from left to right in Figure 2. It is also worth noting that the minimal polynomial of Q(0, 0) has genus zero (so that the corresponding curve has a rational parametrization), except for the last weighted model, which had never been solved so far: where Z = 1 + 12t + 40t 2 + O(t 3 ) satisfies a quartic equation of genus 1: 27 Z 4 − 18 10000 t 4 + 9000 t 3 + 2600 t 2 + 240 t + 1 Z 2 + 8 10 t 2 + 6 t + 1 102500 t 4 + 73500 t 3 + 14650 t 2 + 510 t − 1 Z = 10000 t 4 + 9000 t 3 + 2600 t 2 + 240 t + 1 2 .

An analytic invariant method
We now move to an analytic world, and consider Q(x, y) ≡ Q(x, y; t) as a function of three complex variables. This section borrows its notation and several important results from the analytic approach of quadrant walks, developed first in a probabilistic framework, and then in an enumerative one [FIM99,Ras12].

Preliminaries
Observing that we see that Q(x, y; t) is analytic in {|t| max(1, |x|) max(1, |y|) < 1/|S|} (at least), and that this domain is a neighborhood of the polydisc {|x| 1, |y| 1, |t| < 1/|S|}. Figure 5. Plot of d(x) for x real: the main two possibilities, depending on the sign of x 4 . Note that x 1 may be non-negative, and x 4 may be +∞.
The roots Y 0,1 of the kernel (now called branches) are where a, b and c are defined in (5). The discriminant d( has degree three. We define symmetrically the branches X 0,1 (y) and their four branch points y 1 , . . . , y 4 . One key difference with the formal framework adopted so far is the following: In this section, t is a fixed real number in (0, 1/|S|). Moreover, we only consider non-singular, unweighted models.
Lemma 17 (Properties of the branch points [Ras12, Sec. 3.2]). The branch points x 's are real and distinct. Two of them (say x 1 and x 2 ) are in the open unit disc, with x 1 < x 2 and x 2 > 0. The other two (say x 3 and x 4 ) are outside the closed unit disc, with x 3 > 0 and Of course, symmetric results hold for the branch points y . and [x 3 , x 4 ], the two branches Y 0,1 still exist and are complex conjugate (but possibly infinite at x 1 = 0 as discussed in the next lemma). At the branch points x i , we have Y 0 (x i ) = Y 1 (x i ) (when finite), and we denote this value by Y (x i ). A key object in our definition of weak invariants is the curve L (depending on t) defined by By construction, it is symmetric with respect to the real axis.
We denote by G L the domain delimited by L and avoiding the real point at +∞. See Figure 6 for examples.
Lemma 18 (Properties of the curve L). The curve L is symmetric in the real axis. It intersects this axis at Y (x 2 ) > 0.
If L is unbounded, Y (x 2 ) is the only intersection point. This occurs if and only if neither (−1, 1) nor (−1, 0) belong to S. In this case, x 1 = 0 and the only point of [x 1 , x 2 ] where at least one branch Y i (x) is infinite is x 1 (and then both branches are infinite there).
Consequently, the point 0 is either in the domain G L or on the curve L. The domain G L also contains the (real) branch points y 1 and y 2 , of modulus less than 1. The other two branch points, y 3 and y 4 , are in the complement of G L ∪ L. The domain G L coincides with the region denoted  Table 3; t = 0.2, 0.25, 0.28, 0.3 and 1/3 = 1/|S| from right to left). The dashed curve is obtained for a value t c 1/|S|, where L stops being smooth, but here we only consider values of t less than 1/|S|.
Proof. This lemma is merely an extension to our enumerative framework of results known to hold in the probabilistic framework of [FIM99]: see in particular Thm. 5.3.3 and its proof.
Since d(x) < 0 in (x 1 , x 2 ), the curve L intersects the real axis at two points at most, namely We begin with the polynomial a(x), which is (at most) quadratic. If a(x) = 0 for some real x, then d(x) = b(x) 2 0, hence the sign of a(x) is constant on the interval (x 1 , x 2 ). Since a(x 2 ) > 0 (because x 2 > 0, see Lemma 17), we also have a(x 1 ) 0. Now consider the polynomial b(x), which is also quadratic at most. We have b(0) 0 and b(1) < 0 (by our choice of t), hence b(x) has one root x b in [0, 1): exactly one, since if b(x) is quadratic, it must have a root larger than 1 because t > 0. Moreover, d( If a(x 1 ) = 0, the condition d(x 1 ) = 0 implies that b(x 1 ) = 0 as well. Hence x 1 coincides with x b , which is non-negative; but a(x b ) = 0 then forces x b = 0. Thus b(0) = 0 = a(0), which is equivalent to saying that neither (−1, 0) nor (−1, 1) belong to S. It is readily checked that in this case each Y i (x) tends to infinity as The limit case Y (x 1 ) = 0 occurs when b(x 1 ) = 0 and c(x 1 ) = 0 (since d(x 1 ) = 0). Hence x 1 coincides again with x b , which is non-negative, and the condition c(x b ) = 0 forces x b = 0. Thus b(0) = 0 = c(0), which is equivalent to saying that neither (−1, 0) nor (−1, −1) belong to S. It is readily checked that in this case Y (0) = 0 indeed.
It follows from the results established so far that the intersection of the domain G L with the real axis is (Y (x 1 ), Y (x 2 )), where by convention Y (x 1 ) = −∞ if L is unbounded. Moreover, either Y (x 1 ) = 0 and thus 0 ∈ L, or 0 ∈ (Y (x 1 ), Y (x 2 )). We now want to prove that Y (x 1 ) < y 1 < y 2 < Y (x 2 ) < y 3 , and y 4 < Y (x 1 ) if y 4 < 0. Let us begin with Y (x 1 ) < y 1 , assuming Y (x 1 ) is finite (otherwise there is nothing to prove). We observe that d(Y (x 1 )) 0: otherwise, the roots of K(x, Y (x 1 )) would be complex conjugate or infinite, while one of them is x = x 1 .
Hence Y (x 1 ) cannot be in any of the intervals (y 1 , y 2 ) or (y 3 , y 4 ). Since it is non-positive, as proved above, it is necessarily less than or equal to y 1 , and larger than or equal to y 4 if y 4 < 0.
Similarly, Y (x 2 ) cannot be in any of the intervals (y 1 , y 2 ) or (y 3 , y 4 ). Since it is positive (as proved above), it is either larger than or equal to y 2 , or in (0, y 1 ]. It remains to exclude the two cases 0 < Y (x 2 ) y 1 and 0 < y 4 Y (x 2 ).
If 0 < Y (x 2 ) y 1 then each function X i is continuous on the interval [Y (x 2 ), y 1 ]. Let X i be the branch of X satisfying X i (Y (x 2 )) = x 2 > 0. Since X i (y 1 ) 0, there exists a real number y ∈ (Y (x 2 ), y 1 ], hence necessarily positive, such that X i (y) = 0. That is, K(0, y) = 0 = c(y), which is impossible for y positive.
The argument excluding the case 0 < y 4 Y (x 2 ) is similar: if fact, replacing the step set S by S := {(i, −j) : (i, j) ∈ S} leaves the x unchanged, replaces the set {y : 1 i 4} by {1/y : 1 i 4}, and finally replaces Y i by 1/Y i . With these remarks at hand, one realizes that if 0 < y 4 Y (x 2 ) for one model, then 0 < Y (x 2 ) y 1 for the reflected one.
We still have to exclude the limit cases where Y (x ) would be one of the branch points y i . This would mean that the system K(x, y) = d(x) = d(y) = 0 has a solution. Writing K(x, y) as in (10), and eliminating x and y between these three equations gives a polynomial in t and the weights w i,j that must vanish. One can check that among the 79 unweighted models (w i,j ∈ {0, 1}), those that cancel this polynomial are exactly the 5 singular models.
Finally, since G L contains y 1 , it must coincide with the component of Among the models having decoupling functions (Tables 3 and 4), the only one for which L goes through the point 0 is model #9 in Table 4. The only one for which L is unbounded is the reverse Kreweras model (second model in Table 3). In fact, the method that we are going to present in this section to solve models having a decoupling function is more elegant when L is bounded: this is why three models in Table 4 differ from the original classification of [BMM10] by an x/y-symmetry (Figure 7). We will still illustrate on the case of reverse Kreweras walks what can be done when L is unbounded. Note that the condition for unboundedness is that K(0, y) has no root (and then it equals t). We now recall some properties of the function S(y) = K(0, y)Q(0, y). It is originally defined around y = 0, and analytic (at least) in the unit disc D. This disc contains the points y 1 and y 2 , and thus intersects the domain G L by Lemma 18.
Proposition 19 (The function S(y)). The function S(y) = K(0, y)Q(0, y) has an analytic continuation in D ∪ G L , with finite limits on L. Moreover, for x ∈ [x 1 , x 2 ] ⊂ (−1, 1) and Note that it follows from (27) that for those values of x, an identity that will be combined with the properties of decoupling functions.
Proof. The first point (analyticity) is Theorem 5 in [Ras12]. In order to prove the other statements, we need a more complete picture of the properties of R and S, which can be found in [Ras12].
Let us define the curve M as the counterpart of L for the branches X i : that is, M = X 0 ([y 1 , y 2 ]) ∪ X 1 ([y 1 , y 2 ]). Define the domain G M as the counterpart of G L . Let X 0 be the branch of X satisfying |X 0 (y)| |X 1 (y)| for all y ∈ C (see [Ras12, Lem. 1]), and define Y 0 (x) analogously. Then X 0 is a conformal map from G L \ [y 1 , Moreover, it is shown in the proof of [Ras12, Thm. 5] that • R has an analytic continuation on the domain D ∪ G M , which is included in D ∪ {x : |Y 0 (x)| < 1}, • symmetrically, S has an analytic continuation on the domain D ∪ G L , which is included in D ∪ {y : |X 0 (y)| < 1}, • with these continuations, the following identity holds on y ∈ D ∪ G L : With these results at hand, let us now prove that S has finite limits on L. Take y 0 ∈ L.
Let y tend to y 0 in G L . We can write y = Y 0 (x), where x ∈ G M tends to x 0 (c'est tout à fait clair, cette limite ?). Given that X 0 and Y 0 are inverse maps, (29) reads so that, as x tends to x 0 and y to y 0 , Hence S has finite limits on L. Denoting the right-hand side by S(y 0 ), this also establishes (27), since we can take for y 0 any It remains to prove that S is bounded on G L ∪ L. If G L is bounded, there is nothing more to prove. Otherwise, we know from Lemma 18 that neither (−1, 1) nor (−1, 0) are in S. Then (−1, −1) and (0, 1) must be in S, and it is easy to check that one of the branches X(y) is asymptotic to −1/y 2 as y → −∞, while the other tends either to a non-zero constant, or to infinity. Since X 0 is defined to be the "small" branch, we conclude that X 0 (y) ∼ −1/y 2 at infinity. Returning to (29), this implies that S(y) tends to 0 as y tends to infinity in G L , and completes the proof of the proposition.

Weak invariants
Definition 20. A function I(y) is a weak invariant of a quadrant model S if: • it is meromorphic in the domain G L , and admits finite limit values on the curve L, • for any y ∈ L, we have I(y) = I(y), where now the bar denotes the complex conjugate.
The second condition also reads I(Y 0 ) = I(Y 1 ) for x ∈ [x 1 , x 2 ], because two conjugate points y and y of the curve L are the (complex conjugate) roots of K(x, y) = 0, for some x ∈ [x 1 , x 2 ]. This condition is thus indeed a weak form of the invariant condition of Lemma 5. Hence, if the model admits a rational invariant I(y) in the sense of Lemma 5, having no pole on L, then I is also a weak invariant. However, the above definition is less demanding, and it turns out that every non-singular quadrant model admits a (non-trivial) weak invariant, which we now describe.
This invariant, traditionally denoted w(y) in the analytic approach to quadrant problems [FIM99,Ras12], is in addition injective in G L . In analytic terms, this third condition makes it a conformal gluing function for the domain G L . Explicit expressions of conformal gluing functions are known in a number of cases (when the domain is an ellipse, a polygon, etc.). In our case the bounding curve L is a quartic curve, and w can be expressed in terms of Weierstrass' elliptic functions (see [FIM99,Sec. 5.5.2.1] or [Ras12, Thm. 6]; note that in our paper we exchange the roles played by x and y in these two references): where the various ingredients of this expression are as follows. First, f (y) is a simple rational function of y whose coefficients are algebraic functions of t: where the y 's are the branch points of the functions X 0,1 and d(y) is the counterpart for the variable y of the discriminant d(x) (so that d(y 4 ) = 0). The next ingredient is Weierstrass' elliptic function ℘, with periods ω 1 and ω 2 : Then ℘ 1,2 (z) (resp. ℘ 1,3 (z)) is the Weierstrass function with periods ω 1 and ω 2 (resp. ω 1 and ω 3 ) defined by: , , These definitions make sense thanks to the properties of the y i 's and Y (x i )'s (see Lemmas 17 and 18). If Y (x 1 ) is infinite (which happens if and only if neither (−1, 0) nor (−1, 1) are in S), the integral defining ω 3 starts at −∞. Note that ω 1 ∈ iR + and ω 2 , ω 3 ∈ R + . Finally, as the Weierstrass function is not injective on C, we need to clarify our definition of ℘ −1 1,2 in (30). The function ℘ 1,2 is two-to-one on the fundamental parallelogram [0, ω 1 ) + [0, ω 2 ) (because ℘(z) = ℘(−z + ω 1 + ω 2 )), but is one-to-one when restricted to a half-parallelogrammore precisely, when restricted to the open rectangle (0, ω 1 ) + (0, ω 2 /2) together with the three boundary segments [0, ω 1 /2], [0, ω 2 /2] and ω 2 /2 + [0, ω 1 /2]. We choose the determination of ℘ −1 1,2 in this set.
Proposition 21 (The function w(y)). The function w defined by (30) is a weak invariant, in the sense of Definition 20. It is moreover injective on G L , and has in this domain a unique (and simple) pole, located at y 2 . The function w admits a meromorphic continuation on C \ [y 3 , y 4 ].
In fact, w(y) is a rational function of y if S is one of the 23 models with a finite group, except for the 4 algebraic models (Figure 2, left), where it is algebraic (see [Ras12, Thm. 2 and 3]). We refer to Section 8.1 for a further discussion of the connection between the weak invariant w and the rational invariant J in the finite group case. In the infinite group case, w(y) is not algebraic, nor even D-finite w.r.t. to y, see [Ras12, Thm. 2]. However, we will prove in Theorem 33 that it is D-algebraic in y, and also in t.

The analytic invariant lemma -Application to quadrant walks
We now come with an analytic counterpart of Lemma 2, which applies to the weak invariants of Definition 20.
Lemma 22 (The analytic invariant lemma). Let S be a non-singular quadrant model and A(y) a weak invariant for this model. If A has no pole in G L (and, in the case of a non-bounded curve L, if A is in addition bounded at ∞), then it is independent of y.
Our main result tells that, for each decoupled model with an infinite group (Table 4), the series Q(0, y) has a rational expression in terms of t, y, the function w(y) and some of its specializations. Moreover, this expression is uniform for the first 8 models of the table. The 9th one stands apart, and this is related to the fact, noted after Lemma 18, that the curve L contains the point 0 in this case; equivalently, K(0, y) = ty 2 . Table 7. Values of p, r, α and g 0 in Theorem 23. We denote j = e 2iπ/3 .
Theorem 23. Let S be one of the first 8 models of Table 4, with associated decoupling function G(y). Let p be the unique pole of G, and let r be the residue of G(y) at y = p. Finally, let α be any root of K(0, y), and let g 0 be the constant term of G(y) in its expansion around y = α.
Then the pole p belongs to the domain G L , the point α belongs to G L ∪ L and for y ∈ G L , the series S(y) = K(0, y)Q(0, y) is given by: where w(y) is the weak invariant defined by (30). The values of p, r, α and g 0 are made explicit in Table 7. For instance, for models #1 and #6, which have decoupling function G(y) = −1/y, while for models #3 and #8, with decoupling function G(y) = −y − 1/y, one has: For the 9th model, where G(y) is given in Table 4.

Remarks
1. The expression of S(y) is the same in cases #1 and #6, and in cases #3 and #8 as well, but of course the value of w(y) depends on the details of the model.

2.
For models #2 and #7, Table 7 shows that we have a choice for the value of α: for the first model α = ±i, for the second one α = j or j 2 . But one easily checks that g 0 is the same (namely 0, resp. 1) for both choices of α, and moreover w takes the same value at both points α. This comes from the fact that the two possible values of α are the (complex conjugate) values Y 0 (0) and Y 1 (0), and that w(y) is an invariant. Hence both choices of α give the same expression of S(y).
3. The theorem states that our expressions of S(y) are valid in G L . But combined with Proposition 21, they imply that S can be meromorphically continued to the whole of C \ [y 3 , y 4 ]. 4. The above expressions of S(y) differ from those obtained in the past using the complex analytic approach of [Ras12] by the fact that they do not involve any integration. This opens the way to D-algebraicity, as proved in the next section. The connection between our expressions and the earlier ones is discussed further in Section 8.2.
We begin with a separate lemma dealing with the location of α. The case α = 0 being already addressed in Lemma 18, we focus here on the other cases.
Proof of Theorem 23. Let S be one of the first eight models of Table 4, and G(y) the associated decoupling function. By Lemma 9, the identity holds at the level of formal power series (in t, with rational coefficients in x). Returning to the analytic framework where t is fixed in (0, 1/|S|), this identity holds for any x ∈  Consider now the function − rw (p) w(y)−w(p) , where r is the residue of G at p. By Proposition 21, this is also a weak invariant, with a single pole in G L , found at y = p (note that w (p) cannot vanish since w is injective on G L ). Its residue at p is −r. Then the invariant lemma (Lemma 22) implies that this function differs from L(y) by a constant c: .
Since both functions have finite limits of L, this identity holds on L as well. To conclude the proof of (34), it suffices to determine the constant c. Let α be any root of K(0, y) (see Table 7). Since |α| 1, Q(0, y) and S(y) = K(0, y)Q(0, y) are analytic in a neighborhood of α, as explained at the beginning of this section, and S vanishes at this point. It remains to expand the above identity at y = α, up to the order O(y − α), to determine the value of c as given in the theorem.
We now examine the ninth model, which differs from the first eight for two (related) reasons. First, its decoupling function G still has a unique pole (at p = 0) but this pole has order two (see Table 4). Moreover, the curve L goes through 0 (Lemma 18). However, the idea of the proof is the same as for the first eight models: we will prove that 1/(w(y) − w(p)) also has a double pole at 0, and that subtracting a multiple of this function from S(y) − G(y) yields a pole-free invariant -which is unexpected as there might remain a pole of order 1.
So let us examine the function w(y) near y = 0. Proposition 21 implies that w is analytic in a neighborhood of 0 ∈ L. Let us write w(y) = k 0 w k y k in this neighborhood. Solving K(x, y) = 0 in the neighborhood of x = 0 gives for Y 0 and Y 1 the following expansions, valid in [0, x 2 ] (recall that x 1 = 0): x 2 ], which forces w 1 = 0, w 3 = w 2 /t, and further identities relating the coefficients w k . Moreover, w 2 = 0: given the form w(y) = ℘(Z(y)) of (30), and the fact that Z (0) cannot vanish (this follows from the identity (49) proved below in Section 6), having w 1 = w 2 = 0 would mean that ℘ has a multiple root, namely Z(0), which is never true for a Weierstrass function. Hence, around y = 0 we have: Let us now compare this to the expansion of S(y) − G(y) near y = 0, recalling that S(0) = 0: This shows that S(y) − G(y) − w 2 /(w(y) − w(0)) is an invariant with no pole on G L . Applying Lemma 22 implies that this function is constant in G L , and the above expansions give this value as w 4 /w 2 − 1/t 2 , as stated in (35).
The finite group case. Our approach using weak invariants is robust, and applies in a similar fashion to models with a finite group, as long as the kernel K(0, y) has at least one root α. Indeed, we have used α above to identify the constant c. As explained below Lemma 18, this is equivalent to saying that the curve L is bounded, which holds for Kreweras' model, the double Kreweras' model and Gessel's model (Figure 2). In all three cases, the function S(y) is still given by (34), where the decoupling function G is given in Table 3, and the various constants by Table 8. Table 8. Values of p, r, α and g 0 for three algebraic models.
We expect similar formulas to hold for the four weighted algebraic models of Figure 2, right, provided one develops the counterpart of [Ras12] for steps with (real and positive) weights.
Let us now examine what happens in a model for which K(0, y) = t, by considering the reverse Kreweras model (Figure 6, right). We follow the proof of Theorem 23. By Proposition 19, the function L(y) = S(y) − G(y) = tQ(0, y) + 1/y is meromorphic in G L , with a unique pole at 0, and is bounded at infinity. It is thus an invariant. The analytic invariant lemma tells us that for some constant c. Since K(0, y) = t has no root (in y), we cannot use the same trick as in the proof of Theorem 23 to determine c. But we can expand the above identity at y = 0 to obtain tQ(0, 0) = S(0) = − w (0) 2w (0) + c, and then at the unique point y c ∈ (0, 1) such that K(y, y) = 0 (Figure 9; this point is always in G L since Y (x 2 ) = 1/ √ x 2 > 1): Now applying (27) with x = y c , and using the x/y symmetry of the model, we find that 2S(y c ) = y 2 c + S(0), which finally gives

Differential algebraicity
As recalled in the introduction, quadrant walks have a D-finite generating function if and only if the associated group is finite -we can now say, if and only if they admit rational invariants (Theorem 7). One outcome of our analytic invariant approach is that models with an infinite group having decoupling functions still satisfy polynomial differential equations.
Theorem 25. For any of the 9 models of Table 4, the generating function Q(x, y; t) is differentially algebraic (or: D-algebraic) in x, y, t. That is, it satisfies three polynomial differential equations with coefficients in Q: one in x, one in y and one in t.

Generalities
We consider an abstract differential field K of characteristic 0, with derivation δ. Typical examples occurring in this section are: • the field of meromorphic functions in k variables x 1 , . . . , x k over a complex domain D of C k , equipped with one or several of the derivations ∂/∂x i , • the quotient field of the (integral) ring of formal power series in the variables x 1 , . . . , x k with coefficients in Q, equipped with one or several of the derivations ∂/∂x i , • at the end of the section, the field of Laurent series in t with rational coefficients in x and y, equipped with the three derivations ∂/∂t, ∂/∂x and ∂/∂y.
When F is a function or a series involving k variables x 1 , . . . , x k , as in the above examples, we say that F is differentially algebraic in x i (or: DA in x i ) if it is ∂/∂x i -algebraic. We say that F is globally DA (or DA, for short) if it is DA in each of its variables.
It may be surprising that we do not allow polynomial coefficients in the definition of DA series/functions. In fact, this would not enlarge the DA class: indeed, imagine for instance that the series (or function) F (x, y) satisfies a non-trivial equation where the derivatives are taken with respect to x. Then differentiating with respect to x gives another differential equation, and we can eliminate y between this equation and the above one to obtain a DE free from y. With one more differentiation, we can similarly construct a DE free from x (and y) and conclude that F is DA in x.
An important subclass of DA series (or functions) consists of differentially finite series (or functions): We say that F is D-finite in x i (for short: DF in x i ) if there exist polynomials P j (x 1 , . . . , x k ) in Q[x 1 , . . . , x k ], for 0 j d, not all zero, such that where the derivatives are taken with respect to x i . We say that F is globally differentially finite (or D-finite, or DF) if it is DF in each x i . Finally, a simple subclass of DF series (or functions) consists of algebraic elements, that is, series or functions satisfying a non-trivial polynomial equation P (x 1 , . . . , x k , F ) = 0 with coefficients in Q.
The notions of D-finite and D-algebraic series/functions are standard [Lip89,Lip88,Rit50,Sta99], but D-finite series, having a lot of structure, seem to be discussed more often than DA series, at least in the combinatorics literature. Note that if a series is DF (resp. DA), the function that it defines in (say) its polydisc of convergence is also DF (resp. DA). Conversely, any differential equation satisfied in the neighborhood of some point a = (a 1 , . . . , a k ) by a function F analytic around a holds at the level of power series for the series expansion of F around a.
We will use a number of closure properties. Some of them can be stated in the context of an abstract differential field, using the following proposition.
Proposition 27. Let K be a differential field of characteristic 0, with derivation δ. Let F ∈ K. The following statements are equivalent: (1) F is δ-algebraic, (2) there exists d ∈ N such that all δ-derivatives of F belong to Q(F, δF, . . . , δ (d) F ), (3) there exists a field extension K of Q of finite transcendence degree that contains F and all its δ-derivatives, Proof. 1 ⇒ 2. Take a DE for F of minimal order, and minimal total degree among DEs of minimal order: Applying δ gives: for some polynomials P 1 and P 2 . The total degree of P 1 is less than the total degree of P , and thus by minimality of P , P 1 (F, . . . , δ (d) F ) is non-zero. Property 2 then follows by induction on the order of the derivative.
The following closure properties easily follow.
Corollary 28. The set of δ-algebraic elements of K forms a field. This field is closed under δ, and in fact under any derivation ∂ that commutes with δ.
Proof. Assume that F and G are δ-algebraic. Say that all derivatives of F belong to Q(F, . . . , δ (d) F ), and all derivatives of G belong to Q(G, . . . , δ (e) G). Then all derivatives of F + G and F G belong to Q(F, . . . , δ (d) F, G, . . . , δ (e) G), so that F + G and F G are δ-algebraic by Proposition 27(3). Similarly, all derivatives of 1/F belong to Q(F, . . . , δ (d) F ), so that 1/F is δ-algebraic. The closure under δ of the field of δ-algebraic elements is obvious by Proposition 27(2). Finally, if ∂ is another derivation commuting with δ, then ∂F satisfies the same DE as F , and is thus δ-algebraic.
Specialized to series or functions in k variables x 1 , . . . , x k , the above corollary implies that F + G, F G, 1/F , ∂F/∂x i are DA as soon as F and G are DA. We will need one final closure property, involving composition.
Proposition 29. If F (y 1 , . . . , y r ) is a DA series (or function) of r variables, G 1 (x 1 , . . . , x k ), . . ., G r (x 1 , . . . , x k ) are DA in all x i 's, and the composition H : = F (G 1 , . . . , G r ) is well defined, then H is DA in the x i 's.
Proof. Let us prove that H is DA in x 1 . If, for 1 i r, all y i -derivatives of F (y 1 , . . . , y r ) can be expressed rationally in terms of the first d i derivatives, and all x 1 -derivatives of G j in terms of the first e j ones, then all x 1 -derivatives of H can be expressed rationally in terms of • the functions ∂ a G j /∂x a 1 , for 1 j r and 0 a < e j , • and the functions ∂ c1+···+c k F ∂y c1 1 · · · ∂y cr r (G 1 , . . . , G k ) for 0 c i < d i , 1 i r. We then apply Proposition 27(3).
D-finiteness is not preserved in general by composition, but we still have the following result (see [Lip89] for a proof in the series setting).
Proposition 30. If F (y 1 , . . . , y r ) is a D-finite series (or function) of r variables, G 1 (x 1 , . . . , x k ), . . . , G r (x 1 , . . . , x k ) are algebraic in the x i 's, and the composition H : = F (G 1 , . . . , G r ) is well defined, then H is D-finite in the x i 's.

The Weierstrass elliptic function
It is well known that the Weierstrass function ℘(z, ω 1 , ω 2 ) defined by (32) is DA in z. What may be less known is that it is DA in its periods ω 1 and ω 2 as well. This is what we establish in this section, which is independent of the rest of the paper. We refer to [JS87,WW62] for generalities on the Weierstrass function (but we draw the attention of the reader on the fact that the periods are 2ω 1 and 2ω 2 in [WW62], instead of ω 1 and ω 2 (or ω 3 ) in our paper). Without loss of generality, we assume that (ω 1 /ω 2 ) > 0.
As a function of z, the Weierstrass function ℘ it is meromorphic in C, with poles at all points of the lattice Λ = {iω 1 + jω 2 : (i, j) ∈ Z 2 }. It is periodic of periods ω 1 and ω 2 , and even.
The following non-linear differential equation holds: which we shorten as where g 2 and g 3 (also called invariants in the elliptic terminology!) are defined by and The roots of the polynomial 4x 3 −g 2 x−g 3 are distinct [JS87, Thm. 3.10.9], so that g 3 2 −27g 2 3 = 0. In fact, ℘ only depends on ω 1 and ω 2 through g 2 and g 3 . That is, if another pair ω 1 , ω 2 gives rise to the same values of g 2 and g 3 , then ℘(z, ω 1 , ω 2 ) = ℘(z, ω 1 , ω 2 ). Conversely, for all complex numbers g 2 and g 3 such that g 3 2 − 27g 2 3 = 0, there exist periods ω 1 and ω 2 (not uniquely defined) such that ℘(z, ω 1 , ω 2 ) is the unique function satisfying (37) and having a pole at 0 [JS87, Cor. 6.5.8]. We find convenient to use the letter P for the Weierstrass function seen as a function of g 2 and g 3 rather than ω 1 and ω 2 : Proposition 31. The function P is DA in z, g 2 and g 3 . The function ℘ is DA in z, ω 1 and ω 2 .
The second part of the proposition will follow from the first one by composition, once the following lemma is proved.
Proof of Proposition 31. It follows from (37) and from the discussion following Definition 26 that P(z, g 2 , g 3 ) is DA in z. Indeed, one can eliminate g 2 and g 3 by two differentiations, and obtain To prove that P is DA in g 2 , we start from the two equations: and where ζ is the Weierstrass zeta-function (see [AS64,Eqs. (18.6.20) and (18.6.22)]). We then take the five following equations: (37) (written with P instead of ℘), its derivative with respect to g 2 , (40), its derivative with respect to g 2 , and finally (41). These five equations relate the three functions P, P g2 , P g2g2 to P z , ζ, ζ g2 and P zg2 . Eliminating the latter four series between our five equations gives a DE of order 2 in g 2 (and degree 9) for P, with polynomial coefficients in z, g 2 and g 3 , from which the differential algebraicity of P in g 2 follows.

The weak invariant w
We now consider a non-singular, unweighted quadrant model. Recall the expression (30) of the weak invariant w(y; t), valid for t ∈ (0, 1/|S|) and y in the complex domain G L , which depends on t (Proposition 21). From now on we will often insist on the dependency in t of our functions, denoting for instance ω 1 (t) rather that ω 1 .
Theorem 33. For any non-singular model, the weak invariant w(y; t) defined by (30) can be extended analytically to a domain of C 2 where it is D-algebraic in y and t.
Recall that w(y; t) = ℘ (Z(y; t), ω 1 (t), ω 3 (t)) , where the periods ω 1 and ω 3 are given by (33) and the first argument of ℘ is We will argue by composition of DA functions. We have already proved in the previous subsection that ℘ is DA in its three arguments. Our next objective will be to prove that ω 1 and ω 3 are DA in t (and in fact D-finite, see Lemma 35). We will then proceed with the bivariate function Z, which is also D-finite (Lemma 36). As a very first step, we consider the branch points x .
Lemma 34. The functions x 1 , x 2 , x 3 and x 4 (when it is finite) are algebraic functions of t. They are analytic and distinct in a neighborhood of the interval (0, 1/|S|). The same holds for the branch points y .
Proof. The x i 's are the roots of d(x) = b(x) 2 − 4a(x)c(x). This is a cubic or quartic polynomial in x, with coefficients in Q[t]. Its discriminant does not vanish in (0, 1/|S|) since the x i 's are distinct on this interval (see Lemma 17). Its dominant coefficient is easily checked to be c t 2 , for some c = 0. Since the singularities of algebraic functions are found among the roots of the discriminant and of the dominant coefficient, we conclude that the x i 's are non-singular on (0, 1/|S|), and thus (since their singularities are isolated) in a complex neighborhood of this segment.
Proof. The periods ω i are expressed in (33) as elliptic integrals. Using the classical reduction to Legendre forms [WW62,Sec. 22.7], we can express them in terms of complete and incomplete elliptic integrals of the first kind, defined respectively, for k ∈ (−1, 1) and v ∈ (−1, 1), by Then we claim that: and where the prefactor α is an algebraic function of t, which depends on the degree (3 or 4) of d(y), and of its dominant coefficient d 3 or d 4 : otherwise.
To obtain the above expressions of the periods, one starts from their original expressions in terms of d(y) (see (33)) and performs the following change of variable in the integrand (for ω 1 , ω 2 and ω 3 respectively): The calculation is then straightforward.
The function K(k) has a convergent expansion of radius 1 around k = 0: It is D-finite, with differential equation Its only singularities are at ±1, and it has an analytic continuation of C \ ((−∞, −1) ∪ (1, +∞)). By Lemma 34, the arguments involved in the expressions (44) of ω 1 and ω 2 still have modulus less than 1 in some neighborhood of (0, 1/|S|), where the ω i are thus analytic. By Proposition 30, these two periods are also D-finite in t.
Let us now return to the expression (45) of ω 3 . The function F (v, k) has an expansion around (0, 0) that converges absolutely for |v| < 1 and |k| < 1: It is D-finite in each of its two variables (as a bivariate series and thus as a function). Indeed, ∂F ∂v and ∂ 3 F ∂k 3 = 0. Again, we conclude that ω 3 is D-finite in a neighborhood of (0, 1/|S|) by composition with algebraic functions.
Lemma 36. The function Z(y; t) defined by (42) for t ∈ (0, 1/|S|) and y ∈ G L can be analytically continued to a domain of C 2 in which it is D-finite in t and y.
Let us now differentiate Z with respect to y: Upon squaring this identity, and using first (48), and then (47), we obtain by definition of f . If y 4 is infinite, that is, d 4 = 0, then the parametrization of K(x, y) = 0 is and the identity (Z (y)) 2 = 1/(4 d (y)) still holds. Another property of the parametrization of the kernel by ℘ 1,2 is that f (y 2 ) = ℘ 1,2 ( ω1+ω2 2 ) (see [Ras12], below (18), recalling that we have swapped the roles of x and y). Hence, given our convention in the definition of ℘ −1 1,2 in Section 5.2, we have Z(y 2 ; t) = 0.
(The minus sign comes again from the determination of ℘ −1 1,2 that we have chosen, which has real part in [0, ω 2 /2]. Hence the real part of Z(y; t) is non-positive.) This integral can be expressed in terms of the incomplete elliptic function F (v, k) defined by (43), as we did for the period ω 3 in the proof of Lemma 35: where the prefactor α is given by (46) and the second argument of F is y3−y1 y3−y2 if y 4 is infinite. Since F is D-finite, and its arguments algebraic in t, we conclude once again by a composition argument.

The generating function Q(x, y; t) of decoupled quadrant walks
We now return to the 9 models with an infinite group for which we have obtained a rational expression of Q(0, y; t) in terms of the weak invariant w(y; t) (Theorem 23). We want to prove that the series Q(x, y; t) is D-algebraic (in t, x and y) for each of them, as claimed in Theorem 25.
Let us first prove that Q(0, y; t) is DA. Theorem 23 gives an expression of S(y; t) := K(0, y; t)Q(0, y; t) in terms of the weak invariant w(y; t), valid for t ∈ (0, 1/|S|) and y in G L . By Theorem 33, the weak invariant has an analytic continuation on a complex domain, where it is DA. The closure properties of Propositions 28 and 29 then imply that S(y; t) is also DA, first as a meromorphic function of y and t, then as a series in these variables. The same then holds for Q(0, y; t).
Let us now go back to Q(x, 0; t), using where Y 0 (x; t) is the root of the kernel that is a power series in t (with coefficients in Q[x,x]). Again, we conclude that Q(x, 0; t) is DA using the closure properties of Propositions 28 and 29 (since Y 0 is a series in t with coefficients in Q[x,x], this is where we take Q(x)((t)) as our differential field, as discussed at the beginning of Section 6.1). A final application of Proposition 28, applied to the main functional equation (4), leads us to conclude that Q(x, y; t) is DA as a three-variate series.
6.5. Explicit differential equations in y We now explain how to construct, for the 9 models of Table 4, an explicit DE in y satisfied by the series Q(0, y) ≡ Q(0, y; t). This DE has polynomial coefficients in t and y. Depending on the model, the order of this DE ranges from 3 to 5. We do not claim that it is minimal. The 9 DEs thus obtained have been checked numerically by expanding Q(0, y) in t up to order 30. The corresponding Maple session is available on the authors' webpages. The construction of explicit DEs in t seems more difficult, as discussed later in Section 8.3.
We start from the expression of S(y) = K(0, y)Q(0, y) given by Theorem 23, which can be written as: for α, β and γ depending on t only. The weak invariant satisfies a first order DE, derived from (37) and (49): Here, g 2 ≡ g 2 (ω 1 , ω 3 ) and g 3 ≡ g 3 (ω 1 , ω 3 ) depend (only) on t. Upon solving (50) for w(y), (51) gives a first order DE for Q(0, y), the coefficients of which are polynomials in t, y, α, β, γ, g 2 and g 3 . By expanding this DE around y = 0, we obtain algebraic relations between the 5 unknown series α, β, γ, g 2 , g 3 and the series Q 0,i := 1 i! ∂ i Q/∂y i (0, 0) that count walks ending at (0, i), for 0 i m − 1 (where m depends on the model). We then eliminate α, β, γ, g 2 , g 3 to obtain a DE in y that only involves Q(0, y) and the Q 0,i , for 0 i m − 1. For instance, for model #4, we obtain a DE with coefficients in Q[t, y, Q 0,0 , Q 0,1 ] (hence m = 2), while for model #6, the first 4 series Q 0,i are involved (hence m = 4). Note that this DE is informative: expanding it further around y = 0 allows one to relate the series Q 0,i for i m to those with smaller index. For instance, for model #4 we find: Two remarks are in order, regarding models #5 and #9. For model #5, the decoupling function G(y) is singular at y = −1 (rather than y = 0 for the other models), which leads us to write the equation in terms of the y-derivatives of Q(0, y) at y = −1 rather than y = 0. For model #9, a simplification occurs, since Q 0,0 = 1 + tQ 0,1 (due to the choice of steps), and only two derivatives of Q(0, y), namely Q 0,1 and Q 0,2 , occur in the equation. At this stage, we can proceed as described below Definition 26 to eliminate from the equation the series Q 0,i (or ∂ i Q/∂y i (0, −1) for model #5). If m of them actually occur, then the order of the final DE (with coefficients in Q[y, t]) will be m + 1. For model #4 for instance, for which m = 2, we find the following third order DE: Needless to say, we have no combinatorial understanding of this identity. The orders of the DE obtained for the 9 decoupled models are as follows: We have proved in the previous sections that when the function xy is decoupled (in the sense of Definition 11), the nature of the series Q(x, y; t) that counts quadrant walks starting at (0, 0) tends to be simpler: algebraic when the group G(S) is finite, D-algebraic otherwise. In this section, we explore the existence of decoupling functions for other starting points. We expect similar implications in terms of the nature of the associated generating function (but we have not worked this out). Remarkably, we find that some infinite group models that are not decoupled for walks starting at (0, 0) are still decoupled for other starting points -and we thus expect the associated generating function to be D-algebraic. For a given model S, and a, b ∈ N, we denote by q a,b (i, j; n) the number of walks in N 2 with steps in S starting at (a, b) and ending at (i, j). We define the generating function of walks starting at (a, b) by: This series satisfies the following generalization of (4): This leads us to ask for which models S and which values of a and b the function H(x, y) := x a+1 y b+1 is decoupled. We first give a complete answer in the finite group case (Proposition 37). Then we give what we believe to be the complete list of decoupled cases for infinite groups (Proposition 38). We conclude in Proposition 39 with the 4 weighted models of Figure 2. (1) If S is none of the models of Figure  Proof. Recall from Theorem 12 that H(x, y) is decoupled if and only if H α (x, y) = 0, where α = γ∈G(S) sign(γ)γ. We refer to [BMM10, Tables 1-3] for the explicit description of the group G(S). We will use the following notation: for a Laurent polynomial P in a variable z, we denote by [z > ]P (resp. [z < ]P ) the sum of monomials of positive (resp. negative) exponents in z. We call it the positive (resp. negative) part of P in z.
Let us first consider Gessel's model. The group G(S) has order 8 and It is easy to check that if a = b or a = 2b + 1, then H α (x, y) = 0. Conversely, y) is not decoupled. This proves Claim (3).
Claims (1) and (2) are proved in a similar fashion. For instance, for the 16 models having a vertical symmetry, where as before a(x) = [y 2 ]K(x, y) and c(x) = [y 0 ]K(x, y). Thus H α (x, y) is a Laurent polynomial in y, with positive part H(x, y) − H (x, y), and finally, showing that H(x, y) is never decoupled.
One can also give explicit decoupling functions for the four algebraic models: upon generalizing Lemma 9 to the function H(x, y) = x a+1 y a+1 , we can check that all four models admit F (x) = −x −a−1 as x-decoupling function. Similarly, for Gessel's model with starting point (2b + 1, b), a y-decoupling function is G(y) = −y −b−1 . (1, 0) Table 9. Exhaustive list of decoupled cases among models with a finite group (left) and among the 4 weighted models of Figure 2.
Remarks 1. We recall from [BMM10,Prop. 8] that the 19 models that never decouple (case (1) above) can be solved by extracting the positive part (in x and y) in the alternating sum Q α (x, y), where Q(x, y) = xyQ(x, y). Indeed, this positive part turns out to be simply xyQ(x, y). This property is closely related to the above extraction procedure, and to the non-existence of decoupling functions.
2. Given a step set S, one can also ask whether a linear combination a,b c a,b Q a,b (x, y) is (D-)algebraic. This makes sense for instance in a probabilistic setting, where the c a,b 's describe an initial law for the starting point. Again, we expect this to be equivalent to the polynomial H(x, y) := a,b c a,b x a+1 y b+1 being decoupled. We can extend the proof of Proposition 37 to study this question. If S is one of the 19 models listed in (1), then H(x, y) is never decoupled. If S is one of the Kreweras-like models, then H(x, y) is decoupled if and only if c a,b = c b,a for all (a, b). For instance, we expect Q 0,1 + Q 1,0 to be algebraic. The condition is a bit more complex in Gessel's case. 3. As discussed above, the existence of a decoupling function for a finite group model does not imply algebraicity in a completely automatic fashion, and further work is required to prove it. We have done this for Kreweras' walks starting anywhere on the diagonal: the associated generating function, which involves one more variable recording the position of the starting point, is indeed still algebraic.

Models with an infinite group
We now address models with an infinite group, and exhibit decoupling functions in a number of cases. Remarkably, we find that three models that are not decoupled for walks starting at (0, 0) still admit decoupling functions for other starting points. This contrasts with the finite group case.
Proposition 38. Let S be one of the 12 models with an infinite group shown in Table 10. Then the function x a+1 y b+1 is decoupled for the values of (a, b) shown in the corresponding column.
Based on an (incomplete) argument and a systematic search (for a, b 10), we believe these values of (S, a, b) to be the only decoupled cases (for infinite groups).
Proof. Consider a model with kernel K(x, y), and take a rational function H(x, y). Lemma 9 can be readily extended to show that the following conditions are equivalent: (a) H(x, y) is decoupled, that is, there exist rational functions F (x), G(y) such that K(x, y) divides H(x, y) − F (x) − G(y), (b) there exists a rational function F (x) such that H(X 0 , y) − F (X 0 ) = H(X 1 , y) − F (X 1 ), where X 0 , X 1 are the roots of K(x, y) (when solved for x), (c) there exists a rational function G(y) such that where Y 0 , Y 1 are the roots of K(x, y) (when solved for y). We call x-decoupling function (resp. y-decoupling function) of H(x, y) a rational function F (x) (resp. G(y)) satisfying Condition (b) (resp. Condition (c)).
We begin with the starting point (a, a). When a = 0, we have proved that the 9 decoupled models with an infinite group are those numbered from #1 to #9. Now let a be arbitrary, and denote H(x, y) = x a+1 y a+1 . For models #3 and #5, we have 1 X0X1 = y. Hence, This shows that F (x) = −x −a−1 is an x-decoupling function for H(x, y). Similarly, for models #1 and #6, one has 1 Y0Y1 = x, so that G(y) = −y −a−1 is a y-decoupling function for H(x, y). Finally, for a = 1 and model #9, one easily checks that the function is a y-decoupling function for H(x, y).

Weighted models with a finite group
Proposition 39. Consider the four weighted models of Figure 2. The list of starting points (a, b) for which they decouple is given in Table 9. Specializing λ to some complex value in the first model does not yield more decoupled cases.
Proof. We denote as before H(x, y) = x a+1 y b+1 . The first weighted model has a group of order 6. Writing y = z − 1 makes its elements more compact, and we find: Setting z = x in this expression and taking the limit x → ∞ gives For H(x, y) to be decoupled, we need H α (x, x−1) = 0, which forces a = b. Under this assumption we further obtain which now forces a = b = 0. Conversely, if (a, b) = (0, 0) then H(x, y) is decoupled as proved in Section 4.2.
The second weighted model has a group of order 10, and where H(x, y) = H(x, y) − H(y, x). Setting y = x 2 and taking the limit at x → ∞ gives Hence H α (x, y) = 0 implies a = b. Conversely, if a = b, then H(x, y) = 0, so H α (x, y) = 0. The proof for the third model is similar (except that it is easier to expand of H α (x, x 2 ) at x = 0).
Remark. As in the infinite group case, there exist weighted models that do not decouple at

Final comments and questions
We begin here with two comments on our results. In the first one, we relate weak and rational invariants. In the second, we discuss the link between our new expressions of Q(0, y) (Theorem 23) and the integral expressions formerly obtained in [Ras12]. We then go on with a list of open questions and perspectives.

Weak invariants vs. rational invariants in the finite group case
In the finite group case -and in this case only -we were able to exhibit both a rational invariant J(y) (Definition 4) and a weak invariant w(y) (Definition 20). The analytic invariant lemma (Lemma 22) tells us that the weak invariant is in some sense minimal, and thus more intrinsic than the rational invariant. On the other hand, w(y) depends on t and y in a more complex fashion than J(y): indeed, w(y) is known to be algebraic in y (though rational in 19 cases), and D-algebraic in t (by Theorem 33). We still expect it to be algebraic in t, as is known for groups of order 4 (see Eq. (27) in [Ras12]).
In this section, we show how to relate J(y) and w(y) using the analytic invariant lemma.
Let us first consider one of the 16 models with a horizontal symmetry, for which a rational invariant is J(y) := y + 1/y. One can check that the curve L is the unit circle (see [FIM99,Thm. 5.3.3 (i)] for the probabilistic case t = 1/|S|) and in particular the pole y = 0 never lies on L. Hence J(y) is a weak invariant, in the sense of Definition 20. Applying the analytic invariant lemma shows that In particular, we have thus rederived the fact that w(y) is rational in y. We now address the three models that have no horizontal symmetry and have a transcendental generating function (Figure 10). For the first one, J(y) = ty 2 − y − t/y has a single pole at 0.
The curve L is bounded and G L contains 0, so that for some γ that depends on t only. For the second model, J(y) = y t(1+y) 2 + t (1+y) 2 y (as in Gessel's model) has a simple pole at 0 ∈ G L and a double pole at −1 ∈ L ∪ G L . The invariant lemma gives Finally for the third model, J(y) = t/y − yt − (2t + 1)/(y + 1) has a simple pole at 0 ∈ G L and another one at −1 ∈ L ∪ G L , and the previous expression of J in terms of w holds as well.
Let us now address the four algebraic models, for which invariants are given in Table 1. For Kreweras' model, J(y) has a double pole at 0, and the invariant lemma results in: showing that w(y) is quadratic in y.
For reverse Kreweras walks, the curve L is not bounded, and the invariant J(y) = ty 2 −y −t/y is not bounded at infinity. Hence we cannot apply directly Lemma 22. However, it follows from Lemma 40 (proved below) that This shows that w(y) is quadratic in y.
For the double Kreweras model, J(y) has two poles, at 0 and at −1. Both belong to G L , and the invariant lemma results in: showing that w(y) is again quadratic in y.
Finally, for Gessel's model, J(y) has poles at 0 and −1, both belonging to G L , and respectively simple and double. The invariant lemma gives showing that w(y) is (at most) cubic in y.
We conclude this section with the lemma used above for reverse Kreweras walks (see (52)).
Lemma 40. If the curve L is unbounded, then the weak invariant w(y) is analytic at infinity, where the following expansion holds: for some w 2 = 0.

A connection with integral representations of Q(x, y)
Prior to this paper, for a non-singular model with an infinite group, the series Q(x, y) was expressed as a contour integral involving the gluing function w(y) (a.k.a. weak invariant) [Ras12]. If the model has a decoupling function, we have now obtained a simpler, integral-free expression in Theorem 23. We explain here, without giving all details, how to derive it from the integral one, in the analytic setting of Section 5. To avoid technicalities we only consider models such that 0 / ∈ [x 1 , x 2 ], thereby excluding models #2, #7 and #9. Let w(x) be the counterpart of the weak invariant w(y), but for the variable x. In particular, w(x) is a gluing function for the domain G M already introduced in the proof of Proposition 19. Then, for x ∈ G M \ [x 1 , x 2 ], it is known that This is Theorem 1 in [Ras12], stated here with greater precision (indeed, the first term in the integrand is written as Y 0 (u) − Y 1 (u) in [Ras12]). Recall from Subsection 5.1 that the functions Y 0 and Y 1 are not meromorphic on [x 1 , x 2 ], but admit limits from above and below. These limits satisfy More details can be found in the proof of [Ras12, Thm. 1], or in [KR11, Sec. 4] for Gessel's model. Note that the assumption 0 ∈ [x 1 , x 2 ] guarantees that the term w(u) − w(0) does not vanish. When x ∈ [x 1 , x 2 ], and more generally when x is in the unit disk, R(x) is analytic, even though the two terms of (54) are not analytic along this interval. The first crucial point is that we can replace w by w(Y 0 ) in (54), where w is the gluing function (30) for G L . This comes from a combination of three facts: • as demonstrated in [Ras12, Thm. 6], w(Y 0 (x)) is a conformal gluing function for G M , in the sense that is satisfies Proposition 21 -except that we are now in the x variable, and that the pole is located at X(y 2 ) rather than x 2 (note that the invariant property w(Y 0 (x)) = w(Y 1 (x)) spares us the trouble of taking upper or lower limits when defining w(Y 0 (x)) for x ∈ [x 1 , x 2 ]); • any two conformal gluing functions w 1 and w 2 are related by a homography. That is, w 1 = aw2+b cw2+d , for some coefficients a, b, c, d ∈ C (depending on t) such that ad − bc = 0, see [Ras12, Rem. 6]; in the right-hand side of (54) takes the same value, should w be replaced by a w+b c w+d . Now assume that the model admits a (rational) decoupling function G. Then u(Y 0 (u) − Y 1 (u)) = G(Y 0 (u)) − G(Y 1 (u)) (Lemma 9). The integral in (54) thus becomes: In our context, this would mean introducing weights so as to obtain a family of algebraic models converging to a D-algebraic one.
Let us mention another analogy with Tutte's work. Theorem 1 in [KR12] states that for any non-singular infinite group model, there exists a dense set of values t ∈ (0, 1/|S|) such that the generating function Q(x, y; t) is D-finite in x and y. This paper leads us to believe that for decoupled models, this specialization of Q(x, y; t) will even be algebraic. Then Q(x, y) would be algebraic over R(x, y) for infinitely many values of t, while for Tutte's problem, G(1, 0) is algebraic over C(t) for infinitely many values of the parameter q.

Completing the classification of quadrant walks
Given one of the 79 quadrant models, one would like to tell which of the series Q(x, y), Q(0, 0), and Q(1, 1) are algebraic/D-finite/D-algebraic. As far as we know, the yet unsolved questions deal with infinite group models, and with specializations of Q(x, y): • for the 5 singular models, it is known that Q(1, 1) (and hence Q(x, y)) is not Dfinite [MR09,MM14]. What about D-algebraicity? • for the 51 non-singular models for which Q(x, y) is not D-finite, is Q(1, 1) still D-finite?
Is it D-algebraic for more models than those of Table 4? • for these 51 models, Q(0, 0) is not D-finite [BRS14]. But is it D-algebraic for more models than those of Table 4? Another question deals with finite group models: for those that decouple, we have been able to prove algebraicity using a rational invariant and a decoupling function. Is there a way to prove D-finiteness for the others, using only a rational invariant? Is there something like a weak decoupling function?

Other walk models
Could there be an invariant approach for quadrant walks with large steps [FR15,BBMM]? For walks in a higher dimensional cone [BKY16,BBMKM16,DHW16]? For walks avoiding a quadrant [BM16b]? and L(y) = S(y) − G(y), with S(y) = K(0, y)Q(0, y). In each case (except for the reverse Kreweras walks), we construct from J(y) and L(y) a series in t with polynomial coefficients in y satisfying the conditions of Lemma 15. This construction is very similar to what we did in Section 3 for Gessel's model. Applying Lemma 15, and replacing S(y) by its expression in terms of Q(0, y), gives an equation of the form where Pol(x 0 , x 1 , . . . , x k , t, y) is a polynomial with rational coefficients, and A 1 , . . . , A k are k auxiliary series depending on t only (in what follows, they are always derivatives of Q(0, y) with respect to y, evaluated at y = 0 or y = −1). In the case of reverse Kreweras' walks, Lemma 15 is replaced by the substitution-free appproach of Section 4.4, and (57) follows from (26).
We have described in [BMJ06] a strategy to solve equations of the form (57), which we apply successfully in all eight cases. One key point is to decide how many solutions Y ≡ Y (t) the following equation has: and to note that each of them also satisfies ∂ Pol ∂y (Q(0, Y ), A 1 , . . . , A k , t, Y ) = 0.
Each of these series Y is also a double root of the discriminant of Pol with respect to its first variable, evaluated at A 1 , . . . , A k , t, y (and seen as a polynomial in y); see [BMJ06, Thm. 14]. Note that this method does not require to determine the series Y , but only to decide how many such series exist, and, possibly, compute their first few terms. In addition to the original paper [BMJ06], we refer the reader to [BM16a,Sec. 3.4] where an equation of this type, arising in Gessel's model and involving three series A i , is solved. This section is supported by a Maple session available on the authors' webpages, where all calculations are detailed.

A.2. The reverse Kreweras model
This is the model for which we had to develop a substitution-free version of the invariant lemma in Section 4.4. We start from (26), which gives an equation of the form (57): t 2 yQ(y) 2 + (−t 2 yA 1 + ty 3 − y 2 + t)Q(y) − tyA 2 − tA 1 + y 2 = 0, where Q(y) stands for Q(0, y), A 1 is Q(0, 0) ≡ Q(0) and A 2 is Q y (0, 0) ≡ Q (0). Equation (58) has two roots Y + and Y − , which are power series in √ t. Following the approach of [BMJ06, Sec. 7], we write that the discriminant of Pol with respect to its first variable, evaluated at A 1 , A 2 , t, y, has two double roots in y (namely Y + and Y − ). This gives two polynomial equations relating A 1 and A 2 , from which one derives cubic equations for A 1 and A 2 . Both series have a rational expression in terms of the unique series Z ≡ Z(t), with constant term 0, satisfying Z = t(2 + Z 3 ) (this is the same parametrization as in Kreweras' model). The series Z is denoted W in [BMM10, Prop. 14].
Once A 1 and A 2 are known, one recovers Q(0, y) thanks to (60), and this yields the expression of Q(0, y) given in Prop. 14 of [BMM10]. This series has degree 2 over Q(y, Z).
Returning to the original series Q(0, y), which we denote here Q(y), this gives an equation of the form (57), of degree 2 in Q(y), and involving two additional series A 1 = Q(0) and A 2 = Q (0). Equation (58) has two roots, which are power series in √ t. Following the approach of [BMJ06, Sec. 7], we write that the discriminant of Pol with respect to its first variable, evaluated at A 1 , A 2 , t, y, has two double roots in y. This gives two polynomial equations relating A 1 and A 2 , from which one derives quartic equations for A 1 and A 2 . Both series have a rational expression in terms of the unique series Z ≡ Z(t), with constant term 0, satisfying Z(1 − Z) 2 = t(Z 4 − 2Z 3 + 6Z 2 − 2Z + 1).
This series was introduced in [BMM10], where this model was solved for the first time.

A.4. Gessel's model
We start from (15), with the values of a, b, c and d given in Proposition 3. Returning to the original series Q(0, y), which we denote again Q(y), this gives an equation of the form (57), of degree 3 in Q(y), and involving three additional series A 1 = Q(0), A 2 = Q(−1) and A 3 = Q (−1).
Using the first few coefficients of the three series Y , we conclude that indeed Y • = 1, while Y + and Y − satisfy respectively or equivalently, Following the approach of [BMJ06, Sec. 7], we write that the discriminant of Pol with respect to its first variable, evaluated at A 1 , A 2 , A 3 , t, y, has three double roots in y, namely Y • , Y + and Y − . Up to a power of y, this discriminant can be written as a polynomial ∆(s) of degree 6 in s := y + 1/y. The above equations satisfied by the series Y show that ∆(s) vanishes at s = 2, at s = 1/t − 2 and s = −1/t − 2. This gives three polynomial equations relating A 1 , A 2 and A 3 , from which one finally derives a quartic equation for A 2 , and equations of degree 8 for A 1 and A 3 . As in previous papers dealing with Gessel's model, we introduce the quartic series T ≡ T (t) as the unique solution with constant term 1 of T = 1 + 256t 2 T 3 (3 + T ) 3 , and denote Z = √ T = 1 + O(t). Then we find that A 2 = Q(0, −1) = 2 T 3 + T 2 + 27 T + 3 (T + 3) 3 , A 3 = Q y (0, −1) = (Z − 1) Z 8 − Z 7 − 8 Z 5 + 19 Z 4 + 7 Z 3 + 10 Z 2 + 2 Z + 2 Z 3 (Z 2 + 3)
Once A 1 , A 2 and A 3 are known, one returns to the equation that relates them to Q(0, y) (this is essentially (15)). Expressing t 2 and the series A i as rational functions in Z shows that Q(0, y) is cubic over Q(y, Z), and one recovers its expression given in [BM16a, Thm. 1] or [BK10]. It remains to get back to Q(x, 0), which can be done using the equation R(x) + S(Y 0 ) = xY 0 + R(0), where Y 0 is the root of the kernel that is a power series in t. In fact, we prefer to handle Q(xt, 0) rather than Q(x, 0), because it is an even function of t. It is found that Q(xt, 0) has degree 3 over Q(x, Z), and one recovers its expression given in [BM16a, Thm. 1] or [BK10].
Equation (58) has a root Y = 1 + 2λt + O(t 2 ), and the discriminant of Pol with respect to its first variable thus has a double root in y. This gives for A 1 a cubic equation, which can be parametrized rationally by the unique series Z ≡ Z(t), with coefficients in Q(λ) and constant term 0, satisfying: Z(1 + 4Z) = t 1 + 6Z + 12Z 2 + 4(2 + λ)Z 3 .
In particular, the three series Y are cubic. Following the approach of [BMJ06, Sec. 7], we conclude that the discriminant of Pol with respect to its first variable, evaluated at A 1 , A 2 , A 3 , t, y, has three double roots in y, namely Y • , Y + and Y − .