Matrices over polynomial rings approached by commutative algebra

TL;DR

This paper explores the relationship between matrix theory over polynomial rings and commutative algebra, focusing on algebraic invariants, determinantal ideals, and dual varieties, especially for sparse matrices.

Contribution

It introduces new methods linking numerical algebraic invariants with determinantal ideals in polynomial matrix contexts, emphasizing non-generic cases.

Findings

Analysis of dual varieties to determinantal hypersurfaces

Insights into non-generic determinantal ideals

Applications to sparse matrices

Abstract

The main goal of the paper is the discussion of a deeper interaction between matrix theory over polynomial rings over a field and typical methods of commutative algebra and related algebraic geometry. This is intended in the sense of bringing numerical algebraic invariants into the picture of determinantal ideals, with an emphasis on non-generic ones. In particular, there is a strong focus on square sparse matrices and features of the dual variety to a determinantal hypersurface. Though the overall goal is not exhausted here, one provides several environments where the present treatment has a degree of success.