Combinatorial Theory

In this article, we study systems of \(n\geq 1\), not necessarily linear, discrete differential equations (DDEs) of order \(k\geq 1\) with one catalytic variable. We provide a constructive and elementary proof of algebraicity of the solutions of such equations. This part of the present article can be seen as a generalization of the pioneering work by Bousquet-Mélou and Jehanne (2006) who settled down the case \(n=1\). Moreover, we obtain effective bounds for the algebraicity degrees of the solutions and provide an algorithm for computing annihilating polynomials of the algebraic series. Finally, we compare three different strategies for solving systems of DDEs in view of practical applications.

Mathematics Subject Classifications: 05A99, 13A99, 14R15

Keywords: Catalytic variable, algebraic series, formal power series, discrete differential equations, polynomial systems