Divisibility of Exponential Sums and Solvability of Certain Equations Over Finite Fields

Abstract

In [3], Carlitz determined conditions under which infinite families of polynomials have solutions in a finite field. In this paper we extend

Solvability of systems of polynomial equations with some prescribed monomials

“…In this paper we prove that, under some natural conditions, given a system of polynomials F 1 , • • • , F t with monomials of disjoint support, any system F 1 + G 1 , • • • , F t + G t , where the p-weight degree of the G i 's is smaller than the degree of the monomials in the F i 's, is solvable. In particular, we generalize a result of Carlitz [1] and a result of Castro-Rubio-Vega [2] to systems of polynomial equations.…”

“…The condition d divides p − 1 is not needed for q = p in Felszeghy's result. This family of solvable polynomials also satisfies Rédei's Conjecture.In 2008, Castro-Rubio-Vega[2] extended the result of Carlitz with the following theorem, which also satisfies Rédei's Conjecture.Theorem 3 (CRV). Let d i be a divisor of p − 1 and a i ∈ F q * for i = 1, .…”

p-adic Valuation of Exponential Sums in One Variable Associated to Binomials

“…, j N ) providing the minimum value for N i=1 j i , for example, when the associated terms are similar some of them could add to produce higher powers of θ dividing the exponential sum. In [7,8,10], we computed the p-adic valuation of some exponential sums over finite fields for special polynomials. Our results of this paper generalize and improve the results of [8].…”

Construction of systems of polynomial equations with exact p-Divisibility via the covering method

“…In [9] the authors of the present paper generalized Carlitz's result to generalized diagonal equations by computing the exact p-divisibility of the exponential sum of the polynomial.…”