Uniform determinantal representations

TL;DR

This paper introduces the concept of uniform determinantal representations for all polynomials within a certain degree and variable count, providing bounds, constructions, and applications in root-finding and algebraic geometry.

Contribution

It establishes a lower bound on matrix size for uniform representations, presents an optimal construction, and explores their applications and theoretical connections.

Findings

Derived a lower bound on matrix size for uniform representations

Constructed a near-optimal representation matching the lower bound

Improved root-finding techniques for bivariate polynomial systems

Abstract

The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this last area, we introduce the notion of a uniform determinantal representation, not of a single polynomial but rather of all polynomials in a given number of variables and of a given maximal degree. We derive a lower bound on the size of the matrix, and present a construction achieving that lower bound up to a constant factor as the number of variables is fixed and the degree grows. This construction marks an improvement upon a recent construction due to Plestenjak-Hochstenbach, and we investigate the performance of new representations in their root-finding technique for bivariate systems. Furthermore, we relate uniform determinantal representations to vector spaces of singular matrices, and we conclude with a number of future research directions.