Border bases for lattice ideals

TL;DR

This paper develops a method to find all order ideals for lattice ideals, including positive-dimensional cases, and characterizes border bases that are not derived from Gr"obner bases, with explicit results for 2D lattices.

Contribution

It provides a procedure to identify all order ideals of lattice ideals, extending border basis theory beyond zero-dimensional cases and including non-Gr"obner border bases.

Findings

Finite set of possible order ideals for lattice ideals.

Border bases of positive-dimensional lattice ideals can be finitely described.

Not all border bases originate from Gr"obner bases.

Abstract

The main ingredient to construct an O-border basis of an ideal I $\subseteq$ K[x1,. .., xn] is the order ideal O, which is a basis of the K-vector space K[x1,. .., xn]/I. In this paper we give a procedure to find all the possible order ideals associated with a lattice ideal IM (where M is a lattice of Z n). The construction can be applied to ideals of any dimension (not only zero-dimensional) and shows that the possible order ideals are always in a finite number. For lattice ideals of positive dimension we also show that, although a border basis is infinite, it can be defined in finite terms. Furthermore we give an example which proves that not all border bases of a lattice ideal come from Gr\"obner bases. Finally, we give a complete and explicit description of all the border bases for ideals IM in case M is a 2-dimensional lattice contained in Z 2 .