Explicit Determinants of Homogeneous Polynomial Evaluation Matrices and Applications
Somphong Jitman, Wannarut Rungrottheera
TL;DR
The paper addresses determinants of polynomial evaluation matrices A_{a,b}(p) arising from homogeneous bivariate polynomials p of degree k. It develops a Vandermonde-type factorization A_{a,b}(p) = V_{a}^{(k)} D_α (W_{b}^{(k)})^{T}, which immediately yields rank ≤ k+1 and shows det vanish for n ≥ k+2, while enabling an explicit closed form in the border case n = k+1 and a Cauchy–Binet–based expansion for n ≤ k. Key contributions include a complete determinant formula in the border case, a rank-based understanding for general n, and detailed applications such as the sum-form p(x,y)=f(x+y), equivariance under linear changes of variables, and probabilistic non-vanishing bounds over finite fields via Schwartz–Zippel. This unifies and extends classical results for Vandermonde-type matrices and provides practical tools for interpolation, coding theory, and related structured linear-algebra problems. The framework offers a path to analyze non-homogeneous cases and other polynomial forms beyond the homogeneous setting.
Abstract
In this work, the determinants of matrices constructed by evaluating homogeneous bivariate polynomials at pairs of vectors are investigated. For a polynomial $p(x,y)=\sum\limits_{i=0}^k α_i x^{k-i}y^i$, 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.