Walks in the quarter plane: Genus zero case

Abstract

In the present paper, we use Galois theory of difference equations to study the nature of the generating series of (weighted) walks in the quarter plane with genus zero kernel. Using this approach, we are able to prove that the generating series do not satisfy any nontrivial nonlinear algebraic differential equation with rational coefficients.

On the nature of the generating series of walks in the quarter plane

“…Since [DHRS17] and [DHRS18] prove that there is no such relation for any of the 56 walks with infinite group, except the nine differentially algebraic cases of Figure 1, this allowed us to conclude the proof of Theorem 1.…”

“…Theorem 1 generalizes some of results obtained by Melczer and Mishna for walks with genus zero Kernel curves but unfortunately does not allow to retrieve the non holonomy of the excursion series Q(1, 1, t) obtained in [MM14,MR09]. Theorem 1 can be deduced the combination of Theorem 2 below with the d dx (resp d dy )-differential transcendence results of [DHRS18,DHRS17]. Theorem 2.…”

“…Indeed the class of differentially algebraic functions is a very wild class from the analytic point of view and a differential algebraic function might have an infinite number of singularities. Recently, Roques, Singer and the authors of this paper introduced a new approach based on difference Galois theory and algebraic geometry that allowed them to characterize the differential algebraicity of the series in terms of an orbit configuration of certain points of the Kernel curve with respect to the group action (see [DHRS18,DHRS17]). This criteria allowed them to prove that all but 9 of the unweighted walks attached to a genus one curve with infinite group were differentially transcendent with respect to the x and y-variables, and reprove independently from [BBMR16], that the last 9 cases were differentially algebraic with respect to the x and y-variables.…”

“…d dy )-differentially algebraic over Q. The general strategy of the proof of Theorem 2 is similar to the one of [DHRS18,DHRS17] but its implementation required many new ideas. The first major difference with any of the articles quoted above is that we work in a non archimedean setting.…”

“…The Galois theory developed in [HS08] gives Galoisian criteria in terms of the coefficients of a linear difference equation to compute the differential algebraic relations satisfied by the solutions with respect to any set of derivations commuting with the difference operator. Noting that the derivation ∂ s = s d ds commutes with the operator σ q that maps any meromorphic function g(s) onto g(qs), the authors of [DHRS18,DHRS17] applied these Galoisian criteria to deduce the differential transcendence of their archimedean auxiliary function with respect to s. However, the derivation ∂ t = t d dt does not commute with σ q . Our second main contribution is to introduce a convenient Galoisian framework for the t-derivation.…”