The Orbit-Sum Method for Higher Order Equations

TL;DR

This paper extends the orbit-sum method for solving functional equations related to lattice walks, incorporating algebraic and computational techniques to handle larger step sets and formal algebraic extensions.

Contribution

It advances the orbit-sum method by detailing how to perform algebraic computations in splitting fields and interpret elements as series, building on prior algorithms for large steps.

Findings

Develops methods for computations in splitting fields using primitive element theorem.

Provides algorithms for interpreting algebraic elements as series.

Enhances the orbit-sum method for higher order equations with larger steps.

Abstract

The orbit-sum method is an algebraic version of the reflection-principle that was introduced by Bousquet-M\'{e}lou and Mishna to solve functional equations that arise in the enumeration of lattice walks with small steps restricted to $\mathbb{N}^2$. It proceeds by computing a set of algebraic substitutions that can be applied to a given functional equation, forming a linear combination of its transformed versions to the end of eliminating some of the unknowns, and eliminating further unknowns by discarding terms with negative powers. The extension of the orbit-sum method to walks with large steps was started by Bostan, Bousquet-M\'{e}lou and Melczer. They presented an algorithm that computes the minimal polynomials of the algebraic substitutions. We continue their work by explaining, among other things, how to perform computations in their splitting field on the level of ``formal'' algebraic extensions and how its elements can be interpreted as series. We thereby make use of the primitive element theorem, Gr\"{o}bner bases and the shape lemma, and the Newton-Puiseux algorithm.