Module Border Bases

TL;DR

This paper extends the concept of border bases from polynomial ideals to modules and quotient modules, providing algorithms and characterizations for their computation and properties.

Contribution

It introduces the notion of module border bases and quotient module border bases, along with algorithms and criteria for their computation and analysis.

Findings

Existence and uniqueness of module border bases established.

Algorithms for computing module and quotient module border bases developed.

Characterizations and Buchberger's Criterion extended to module border bases.

Abstract

In this paper, we generalize the notion of border bases of zero-dimensional polynomial ideals to the module setting. To this end, we introduce order modules as a generalization of order ideals and module border bases of submodules with finite codimension in a free module as a generalization of border bases of zero-dimensional ideals in the first part of this paper. In particular, we extend the division algorithm for border bases to the module setting, show the existence and uniqueness of module border bases, and characterize module border bases analogously like border bases via the special generation property, border form modules, rewrite rules, commuting matrices, and liftings of border syzygies. Furthermore, we deduce Buchberger's Criterion for Module Border Bases and give an algorithm for the computation of module border bases that uses linear algebra techniques. In the second part, we further generalize the notion of module border bases to quotient modules. We then show the connection between quotient module border bases and special module border bases and deduce characterizations similar to the ones for module border bases. Moreover, we give an algorithm for the computation of quotient module border bases using linear algebra techniques, again. At last, we prove that subideal border bases are isomorphic to special quotient module border bases. This isomorphy immediately yields characterizations and an algorithm for the computation of subideal border bases.