Normal and Triangular Determinantal Representations of Multivariate Polynomials

TL;DR

This paper introduces a new algorithm for converting multivariate polynomials into various determinant forms, potentially enabling more efficient numerical solutions by reducing representation dimensions.

Contribution

It presents a simple algorithm to generate normal, triangular, and reduced determinant representations of multivariate polynomials, improving on existing methods.

Findings

Algorithm produces determinant representations with structured entries.

Enables smaller-dimensional representations for numerical problems.

Applicable to a wide class of multivariate polynomials.

Abstract

In this paper we give a new and simple algorithm to put any multivariate polynomial into a normal determinant form in which each entry has the form , and in each column the same variable appears. We also apply the algorithm to obtain a triangular determinant representation, a reduced determinant representation, and a uniform determinant representation of any multivariable polynomial. The algorithm could be useful for obtaining representations of dimensions smaller than those available up to now to solve numerical problems.