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The Orbit-Sum Method for Higher-Order Equations
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DOI：10.1007/s00026-026-00818-w.png)
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Abstract
En 中文
The orbit-sum method is an algebraic version of the reflection principle that was introduced by Bousquet-M & eacute;lou and Mishna to solve functional equations that arise in the enumeration of lattice walks with small steps restricted to N2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {N}<^>2$$\end{document}. 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 with the goal 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 & eacute;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 & ouml;bner bases and the shape lemma, and the Newton-Puiseux algorithm.
Keywords:
Lattice walks
Generating functions
Functional equations
Orbit-sum method
Journal
A
IF:
0.7
Papers:
45
Citations:
0