Related papers: Exponential sums with sparse polynomials over fini…

We obtain a new upper bound for binary sums with multiplicative characters over variables belong to some sets, having small additive doubling.

We estimate the number of solutions of certain diagonal congruences involving factorials. We use these results to bound exponential sums with products of two factorials $n!m!$ and also derive asymptotic formulas for the number of solutions…

We use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime $p$. In particular, we obtain nontrivial results about the number of solution in boxes with the side…

We extend some methods of bounding exponential sums of the type $\displaystyle\sum_{n\le N}e^{2\pi iag^n/p}$ to deal with the case when $g$ is not necessarily a primitive root. We also show some recent results of Shkredov concerning…

We give a high precision polynomial-time approximation scheme for the supremum of any honest n-variate (n+2)-nomial with a constant term, allowing real exponents as well as real coefficients. Our complexity bounds count field operations and…

Let ${\mathcal H}$ be a multiplicative subgroup of $\mathbb{F}_p^*$ of order $H>p^{1/4}$. We show that $$ \max_{(a,p)=1}\left|\sum_{x\in {\mathcal H}} {\mathbf{\,e}}_p(ax)\right| \le H^{1-31/2880+o(1)}, $$ where ${\mathbf{\,e}}_p(z) =…

We give a simple matrix-based proof of congruence equations modulo a prime $p$ involving sums of binomial coefficients appearing in Pascal's triangle. These equations can be used to construct some groups of exponent $p^n$. These groups, as…

Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime $p$. The sum over the distinct residues can sometimes be computed independent of the prime $p$; for example,…

Let $p(z)=z^n+a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+\ldots+a_1z+a_0$ be a complex polynomial with $a_0\neq 0$ and $n\geq 3$. Several new upper bounds for the moduli of the zeros of $p$ are developed. In particular, if…

We study averages over squarefree moduli of the size of exponential sums with polynomial phases. We prove upper bounds on various moments of such sums, and obtain evidence of un-correlation of exponential sums associated to different…

We give a new bound on the number of collinear triples for two arbitrary subsets of a finite field. This improves on existing results which rely on the Cauchy inequality. We then us this to provide a new bound on trilinear and quadrilinear…

The polynomials of degree $\frac{p-1}{2}$ of range sum $p$ was determined in {\tt arXiv:2311.06136 [math.NT]} for large enough primes. We extend this result by reducing the lower bound for the primes to $23$ by introducing a new and…

We prove prime exponential sums have no better than square root cancellation on average on short intervals, in the sense that $$\frac{1}{x} \sum_{-y< n\le x} \left|\sum_{\substack{n< m \le n+y\\ 1\le m \le x}} \Lambda(m) \mathrm{e}(\alpha…

We bound the tensor ranks of elementary symmetric polynomials, and we give explicit decompositions into powers of linear forms. The bound is attained when the degree is odd.

Our previous theorems on exponential sums often did not apply or did not give sharp results when certain powers of a variable appearing in the polynomial were divisible by p. We remedy that defect in this paper by systematically applying…

We prove new upper bounds for a spectral exponential sum by refining the process by which one evaluates mean values of $L$-functions multiplied by an oscillating function. In particular, we introduce a method which is capable of taking into…

We use an estimate of Aksoy Yazici, Murphy, Rudnev and Shkredov (2016) on the number of solutions of certain equations involving products and differences of sets in prime finite fields to give an explicit upper bound on trilinear…

We estimate weighted character sums with determinants $ad-bc $ of $2\times 2$ matrices modulo a prime $p$ with entries $a,b,c,d $ varying over the interval $ [1,N]$. Our goal is to obtain nontrivial bounds for values of $N$ as small as…

We examine a family of three-dimensional exponential sums with monomials and provide estimates which are in some instances sharper than those stemming from approaches entailing the use of existing bounds pertaining to analogous sums.

In this article, we consider the estimation of exponential sums along the points of the reduction mod $p^{m}$ of a $p$-adic analytic submanifold of $ \mathbb{Z}_{p}^{n}$. More precisely, we extend Igusa's stationary phase method to this…