Exact $p$-divisibility of exponential sums via the covering method

Abstract

In general, the methods to estimate the p-divisibility of exponential sums or the number of solutions of systems of polynomial equations over finite fields are non-elementary. In this paper we present the covering method, an elementary combinatorial method that can be used to compute the exact p-divisibility of exponential sums over a prime field. The results here allow us to compute the exact p-divisibility of exponential sums of new families of polynomials, to unify and improve previously known results, and to construct families of systems of polynomial equations over finite fields that are solvable.

A divisibility approach to the open boundary cases of Cusick-Li-Stǎnicǎ’s conjecture

“…Then, n = 16m − 3 and k = 8m − 4. As m runs through the positive odd numbers, the sequence ν 2 (S(σ 16m−3,8m−4 )) obtains the values 4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7, · · · Compare these values with the ones from ν 2 (S(σ 16m−4,8m−4 )) in Example 3. Note that each term is one more than the corresponding one in the list of the 2-adic valuation of S(σ 16m−4,8m−4 ).…”

“…The first few values of ν 2 (S(σ n,k )), when m runs through the odd positive integers, are given by 5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8, · · · . Again, note that these values are simply the weight of the odd numbers shifted by a constant, which in this case is 4.…”

“…As before, consider the class 16j 4 +9. If a > 2, then the sequence ν 2 (S(σ n,k )) is given by 6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9, · · · This sequence seems to be w 2 (16j 4 + 9) + 4. Observe that now we need a > 2.…”

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 [8] the authors reformulated the concept of coverings of polynomials introduced in [7] to provide an elementary combinatorial method to compute exact pdivisibility of exponential sums over a prime field and an intuitive approach to the construction of systems of polynomial equations that are guaranteed to be solvable. There, the computation of the exact p-divisibility depended on the existence of a unique minimal (p − 1)-covering of the polynomials in the system and the results gave sufficient conditions for its existence.…”