On the power of the discriminant of a univariate polynomial as a certain determinant in positive characteristic

Akira Kurihara

paper

On the power of the discriminant of a univariate polynomial as a certain determinant in positive characteristic

arXiv:2605.29312

Abstract

Let $p$ be a prime. Suppose that integers $r$ , $e$ , $d$ such that $r \geq 2$ , $e \geq 0$ , $0 \leq d \leq p$ are given. Let $f (x) = s_{0} x^{r} + s_{1} x^{r - 1} + \dots + s_{r}$ be a generic polynomial of degree $r$ in characteristic $p$ . We put $f (x)^{e} = \sum_{i \geq 0} c_{i} x^{i}$ . We define a $d \times d$ matrix $M_{d} (f (x)^{e})$ by $M_{d} (f (x)^{e}) = (c_{i p + j - d - 1})_{1 \leq i, j \leq d}$ . In this paper, we shall be concerned with the divisibility of $det M_{d} (f (x)^{e})$ by powers of the discriminant $Δ (f (x))$ of $f (x)$ . First, assuming $s_{0} = 1$ , we study the condition under which $det M_{d} (f (x)^{e})$ is a positive power of $Δ (f (x))$ multiplied by a non-zero constant in $F_{p}$ . Second, for such matrices when $d = r - 1$ , we present a formula for $M_{d} (f (x)^{e})^{- 1} M_{d} (f (x)^{e + 1})$ involving the Bézout matrix of $f^{'} (x)$ and $f (x) - \frac{1}{r} x f^{'} (x)$ . Finally, we present two similar experimental equalities, the first of which involves the determinant $det M_{d} (f (x)^{e})$ .