
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,…
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Taxonomy
TopicsPolynomial and algebraic computation · Commutative Algebra and Its Applications · Cancer Treatment and Pharmacology
