Border Bases for Polynomial Rings over Noetherian Rings

TL;DR

This paper extends the concept of border bases to polynomial rings over Noetherian rings where residue class rings are finitely generated but not necessarily free, providing algorithms and characterizations.

Contribution

It introduces a border division algorithm for a new class of border bases over Noetherian rings and proves their existence and properties.

Findings

Existence of border bases over Noetherian rings with finitely generated residue class rings.

A border division algorithm with proven termination for certain border bases.

Some reduced Gröbner bases are contained within this class of border bases.

Abstract

The theory of border bases for zero-dimensional ideals has attracted several researchers in symbolic computation due to their numerical stability and mathematical elegance. As shown in (Francis & Dukkipati, J. Symb. Comp., 2014), one can extend the concept of border bases over Noetherian rings whenever the corresponding residue class ring is finitely generated and free. In this paper we address the following problem: Can the concept of border basis over Noetherian rings exists for ideals when the corresponding residue class rings are finitely generated but need not necessarily be free modules? We present a border division algorithm and prove the termination of the algorithm for a special class of border bases. We show the existence of such border bases over Noetherian rings and present some characterizations in this regard. We also show that certain reduced Gr\"{o}bner bases over Noetherian rings are contained in this class of border bases.