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

Abstract

Let $p$ be a prime. Suppose that integers $r$, $e$, $d$ such that $r \ge 2$, $e \ge 0$, $0 \le d \le p$ are given. Let $f(x)=s_0 x^r + s_1 x^{r-1} + \cdots + s_r$ be a generic polynomial of degree $r$ in characteristic $p$. We put $f(x)^e=\sum_{i \ge 0} c_i x^i$. We define a $d\times d$ matrix $M_d(f(x)^e)$ by $M_d(f(x)^e) = ( c_{i p + j - d -1})_{1 \le i,\, j \le d}$. 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 ${\mathbb F}_p$. 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 $\det M_d(f(x)^e)$.