Explicit Determinants of Homogeneous Polynomial Evaluation Matrices and Applications

Abstract: In this work, the determinants of matrices constructed by evaluating homogeneous bivariate polynomials at pairs of vectors are investigated. For a polynomial $p(x,y)=\sum\limits_{i=0}^k αi x^{k-i}y^i$, 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.