A note on the regularity of matrices with uniform polynomial entries

TL;DR

This paper investigates the regularity of matrices with uniform polynomial entries, establishing conditions under which they are regular and exploring connections to Vandermonde matrices and Schur polynomials.

Contribution

It introduces new criteria for matrix regularity with polynomial entries and links these to classical algebraic structures like Vandermonde matrices and Schur polynomials.

Findings

Matrices are regular under certain size constraints.

Connections to generalized Vandermonde matrices are established.

Schur polynomials play a key role in the analysis.

Abstract

In this text we study the regularity of matrices with special polynomial entries. Barring some mild conditions we show that these matrices are regular if a natural limit size is not exceeded. The proof draws connections to generalized Vandermonde matrices and Schur polynomials that are discussed in detail.