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Sublogarithmic-Transexponential Series


This thesis is motivated by the open question of whether there are transexponential o-minimal structures. As a candidate for a transexponential o-minimal structure, we suggest $\mathbb{R}_{\mathrm{an},\exp}$ expanded with new symbols for a transexponential function, its derivatives, and their compositional inverses, which we call $\mathbb{R}_{\mathrm{transexp}}$. The main result of the thesis is that the germs at $+\infty$ of $\mathbb{R}_{\mathrm{transexp}}$-terms are ordered and thus form a Hardy field. If $\mathbb{R}_{\mathrm{transexp}}$ is shown to have quantifier elimination in the future, then o-minimality would follow. Chapter 1 provides background on the open problem and more information on $\mathbb{R}_{\mathrm{transexp}}$.

The work of the thesis is to adapt the construction of the field of logarithmic-exponential series in Logarithmic-Exponential Series to build an ordered differential field of sublogarithmic-transexponential series. The sublogarithmic-transexponential series field is constructed to embed the germs of $\mathbb{R}_{\mathrm{transexp}}$-terms, showing that they are ordered. The challenge is to gradually build up the field of series so that we know how the partial structure should be ordered at each stage. We discuss how the ordering quickly becomes complicated even for relatively simple finite sums in Chapter 2.

The construction of the sublogarithmic-transexponential series can be divided into three parts. First, in Chapter 3, we prove that the group ring of certain kinds of finite sums is ordered by giving an algorithm to determine the sign on a sum and showing the algorithm terminates. Next, in Chapter 4, we adapt the construction of the logarithmic-exponential series to build a field of series closed under log, exp, and restricted analytic functions, starting from a field of coefficients and group of monomials satisfying certain assumption. Finally, in Chapter 5, we iterate the construction from Chapter 4 to build the full field of sublogarithmic-transexponential series. We also define a derivation that works like "differentiation with respect to the formal variable of the series."

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