Generating functions for borders

TL;DR

This paper derives a generating function for the index of lattice points relative to finite order ideals, providing a new tool for border bases and solving related difference equations explicitly.

Contribution

It introduces a generating function for the index in border bases, offering a novel explicit solution to certain difference equations.

Findings

Derived the generating function for lattice point indices

Provided explicit solutions to a class of difference equations

Enhanced understanding of border bases and their computational aspects

Abstract

We give the generating function for the index of integer lattice points, relative to a finite order ideal. The index is an important concept in the theory of border bases, an alternative to Gr\"obner bases. Equivalently, we explicitly solve a class of difference equations where the right-hand side is the minimum of a number of affine forms.