On determinantal varieties of Hankel matrices (Q895965)

Summary: Let \(\mathfrak H\) be a class of \(n\times n\) Hankel matrices \(\mathbf{H}_A\) whose entries, depending on a given matrix \(\mathbf A\), are linear forms in \(n\) variables with coefficients in a finite field \(\mathbb F_q\). For every matrix in \(\mathfrak H\), it is shown that the varieties specified by the leading minors of orders from 1 to \(n-1\) have the same number \(q^{n-1}\) of points in \(\mathbb F_q^n\). Further properties are derived, which show that sets of varieties, tied to a given Hankel matrix, resemble a set of hyperplanes as regards the number of points of their intersections.