4 problems

Fix nonnegative integers $x_1,\dots,x_{16}$ with $x_1+\dots+x_8=n$. Let $\mathcal{S}_{\mathbf{x}}$ be the collection of all quadruples $(\mathcal{A}_1,\dots,\mathcal{A}_4)$ whose s…

Let $\mathbb{F}_q$ be a finite field, and let $D_f(\alpha_1,\ldots,\alpha_k)$ be the generalized Vandermonde determinant associated with a polynomial $f$. Punctured-field generaliz…

Let $\mathbb{F}_q$ be a finite field. For a polynomial $f\in\mathbb{F}_q[x]$ and pairwise distinct elements $\alpha_1,\ldots,\alpha_k\in\mathbb{F}_q$, let…

Let $x_1,x_2,\dots,x_n$ be pairwise distinct numbers and let $\omega_1,\dots,\omega_n$ be nonzero. Consider the linear functional … For each $k$, let $H_k$ be the associated Hankel…