Homogeneous Border Bases on Infinite Order Ideals

TL;DR

This paper extends border basis theory from 0-dimensional to positive Krull dimension homogeneous ideals by introducing homogeneous border bases and providing finite verification criteria for their characterization.

Contribution

It introduces homogeneous border bases for infinite order ideals and offers new characterizations, including a finite criterion for formal multiplication matrices.

Findings

Homogeneous border bases are defined for positive Krull dimension ideals.

A characterization via border reductors is provided.

A finite verification criterion for formal multiplication matrices is established.

Abstract

Border bases are traditionally restricted to 0-dimensional ideals due to the finiteness of the underlying order ideal. In this paper we extend the theory to homogeneous ideals of positive Krull dimension by introducing homogeneous border bases, defined relative to an infinite order ideal. Moreover, we provide two characterizations of these bases: one via border reductors and, most notably, one in terms of formal multiplication matrices. Although the latter condition a priori requires verification in infinitely many degrees, we prove that it is sufficient to check only finitely many of them, thereby obtaining an effective criterion.