Toric Border Bases

TL;DR

This paper extends border basis theory to Laurent polynomial systems, introducing a more efficient algorithm for toric roots that directly handles variables and their inverses without additional relations.

Contribution

It develops a new approach for toric border bases that avoids auxiliary variables, characterizes these bases via commutation and inversion relations, and provides a terminating algorithm for zero-dimensional cases.

Findings

Characterization of toric border bases through commutation and inversion relations

Explicit description of the first syzygy module for toric border bases

A new algorithm with proven termination for zero-dimensional toric ideals

Abstract

We extend the theory and the algorithms of Border Bases to systems of Laurent polynomial equations, defining "toric" roots. Instead of introducing new variables and new relations to saturate by the variable inverses, we propose a more efficient approach which works directly with the variables and their inverse. We show that the commutation relations and the inversion relations characterize toric border bases. We explicitly describe the first syzygy module associated to a toric border basis in terms of these relations. Finally, a new border basis algorithm for Laurent polynomials is described and a proof of its termination is given for zero-dimensional toric ideals.