Counting Polynomials with Distinct Zeros in Finite Fields

Let $\mathbb{F}_q$ be a finite field with $q=p^e$ elements, where $p$ is a prime and $e\geq 1$ is an integer. Let $\ell<n$ be two positive integers. Fix a monic polynomial $u(x)=x^n +u_{n-1}x^{n-1}+\cdots +u_{\ell+1}x^{\ell+1} \in \mathbb{F}_q[x]$ of degree $n$ and consider all degree $n$ monic polynomials of the form $$f(x) = u(x) + v_\ell(x), \ v_\ell(x)=a_\ell x^\ell+a_{\ell-1}x^{\ell-1}+\cdots+a_1x+a_0\in \mathbb{F}_q[x].$$ For integer $0\leq k \leq {\rm min}\{n,q\}$, let $N_k(u(x),\ell)$ denote the total number of $v_\ell(x)$ such that $u(x)+v_\ell(x)$ has exactly $k$ distinct roots in $\mathbb{F}_q$, i.e. $$N_k(u(x),\ell)=|\{f(x)=u(x)+v_l(x)\ |\ f(x)\ {\rm has\ exactly}\ k\ {\rm distinct\ zeros\ in}\ \mathbb{F}_q\}|.$$ In this paper, we obtain explicit combinatorial formulae for $N_k(u(x),\ell)$ when $n-\ell$ is small, namely when $n-\ell= 1, 2, 3$. As an application, we define two kinds of Wenger graphs called jumped Wenger graphs and obtain their explicit spectrum.