Counting polynomials with distinct zeros in finite fields.

Let F q be a finite field with q = p e elements, where p is a prime and e ≥ 1 is an integer. Let ℓ , n be two positive integers such that ℓ < n . Fix a monic polynomial u ( x ) = x n + u n − 1 x n − 1 + ⋯ + u ℓ + 1 x ℓ + 1 ∈ F q [ x ] of degree n and consider all degree n monic polynomials of the form f ( x ) = u ( x ) + v ℓ ( x ) , v ℓ ( x ) = a ℓ x ℓ + a ℓ − 1 x ℓ − 1 + ⋯ + a 1 x + a 0 ∈ F q [ x ] . For any non-negative integer k ≤ min { n , q } , let N k ( u ( x ) , ℓ ) denote the total number of v ℓ ( x ) such that u ( x ) + v ℓ ( x ) has exactly k distinct roots in F q , i.e. N k ( u ( x ) , ℓ ) = | { f ( x ) = u ( x ) + v l ( x ) | f ( x ) has exactly k distinct zeros in F q } | . In this paper, we obtain explicit combinatorial formulae for N k ( u ( x ) , ℓ ) when n − ℓ is small, namely when n − ℓ = 1 , 2 , 3 . As an application, we define two kinds of Wenger graphs called jumped Wenger graphs and obtain their explicit spectrum. [ABSTRACT FROM AUTHOR]