On Differentially Algebraic Generating Series for Walks in the Quarter Plane

TL;DR

This paper refines conditions under which the generating series for quarter plane walks are differentially algebraic and provides algorithms to determine this property based on the model's weights.

Contribution

It introduces refined criteria and algorithms using Mordell-Weil lattice theory to identify when the generating series of weighted quarter plane walks are differentially algebraic.

Findings

Refined necessary and sufficient conditions for differential algebraicity.

Algorithms based on Mordell-Weil lattices for weight analysis.

Polynomial conditions on weights for differential algebraicity.

Abstract

We refine necessary and sufficient conditions for the generating series of a weighted model of a quarter plane walk to be differentially algebraic. In addition, we give algorithms based on the theory of Mordell-Weil lattices, that, for each weighted model, yield polynomial conditions on the weights determining this property of the associated generating series.