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

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 $M_d(f(x{)}^e)=(c_{ip+j-d-1}{)}_{1\le i,j\le d}$. In this paper, we shall be concerned with the divisibility of $\det M_d(f(x{)}^e)$ by powers of the discriminant $\mathrm{\Delta }(f(x))$ 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 $\mathrm{\Delta }(f(x))$ 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 $f^{\mathrm{\prime }}(x)$ and $f(x)-\frac{1}{r}x{f}^{\mathrm{\prime }}(x)$. Finally, we present two similar experimental equalities, the first of which involves the determinant $\det M_d(f(x{)}^e)$.