Explicit Determinants of Homogeneous Polynomial Evaluation Matrices and Applications

TL;DR

This paper derives explicit formulas for the determinants of matrices formed by evaluating homogeneous bivariate polynomials at pairs of vectors, revealing conditions for their vanishing and applications in algebra and finite fields.

Contribution

It provides explicit factorization formulas for these determinants, including special cases and connections to Vandermonde determinants, advancing understanding of polynomial evaluation matrices.

Findings

Determinant formulas vanish when matrix size exceeds polynomial degree.

Closed-form determinant involving Vandermonde determinants for the borderline case.

Applications include explicit formulas and bounds over finite fields.

Abstract

In this work, the determinants of matrices constructed by evaluating homogeneous bivariate polynomials at pairs of vectors are investigated. For a polynomial $p(x,y)=\sum\limits_{i=0}^k \alpha_i x^{k-i}y^i$, an explicit factorization of the determinant of the associated $n\times n$ evaluation matrix $A_{\mathbf{a},\mathbf{b}}(p(x,y))=\bigl[p(a_r,b_s)\bigr]_{r,s=1}^n$ is presented for all $n \ge k+1$ and for all pairs of vectors $\mathbf a=(a_1,\dots,a_n)$ and $\mathbf b=(b_1,\dots,b_n)$ of length $n$. In particular, it is proved that $\det (A_{\mathbf{a},\mathbf{b}}(p(x,y)))=0$ when $n \ge k+2$, while in the borderline case $n=k+1$ a closed formula involving Vandermonde determinants is derived in the vector sets and the coefficients of $p(x,y)$. Several well-known determinants, including those arising from $(x+y)^k$ and classical quotient forms $\frac{a^k-b^k}{a-b}$ and $\frac{a^k+b^k}{a+b}$, emerge as special cases. We also provide a discussion for $n \le k$, connecting the problem to symmetric functions and generalized Vandermonde determinants. Finally, applications such matrices and determinants are provided, including an explicit formula and equivariance law under linear changes of variables for the sum-form \(p(x,y)=f(x+y)\), and a non-vanishing bound over finite fields via Schwartz-Zippel lemma.