Related papers: A stratification result for an exponential sum mod…

We discuss exponential sums on affine space from the point of view of Dwork's p-adic cohomology theory

We prove the modulo $p$ and modulo $p^2$ cases of Igusa's conjecture on exponential sums. This conjecture predicts specific uniform bounds in the homogeneous polynomial case of exponential sums modulo $p^m$ when $p$ and $m$ vary. We…

We introduce a general class $F_0$ of additive functions $f$ such that $f(p) = 1$ and prove a tight bound for exponential sums of the form $\sum_{n \le x} f(n) e(\alpha n)$ where $f \in F_0$ and $e(\theta) = \exp(2\pi i \theta)$. Both…

These notes are devoted to the theory of exponential sums over finite fields. The first chapter recalls some of the number-theoretic interest of such sums. The second chapter discusses the $L$-functions attached to such sums, the "Weil…

We examine exponential sums of the form $\sum_{n \le X} w(n) e^{2\pi i\alpha n^k}$, for $k=1,2$, where $\alpha$ satisfies a generalized Diophantine approximation and where $w$ are different arithmetic functions that might be multiplicative,…

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…

Let $\mathbb{F}_{q}$ be a finite field, $\mathbb{F}_{q^s}$ be an extension of $\mathbb{F}_q$, let $f(x)\in \mathbb{F}_q[x]$ be a polynomial of degree $n$ with $\gcd(n,q)=1$. We present a recursive formula for evaluating the exponential sum…

Let $\mathbb{F}_{q}$ be a finite field, $\mathbb{F}_{q^s}$ be an extension of $\mathbb{F}_q$, let $f(x)\in \mathbb{F}_q[x]$ be a polynomial of degree $n$ with $\gcd(n,q)=1$. We present a recursive formula for evaluating the exponential sum…

In the paper we obtain some new upper bounds for exponential sums over multiplicative subgroups G of F^*_p having sizes in the range [p^{c_1}, p^{c_2}], where c_1,c_2 are some absolute constants close to 1/2. As an application we prove that…

By using a variant of Kowalski's large sieve for Frobenius in compatible systems, we obtain zero-density estimates for arguments of $\ell$-adic trace functions over finite fields with values in some algebraic subsets of the cyclotomic…

We use the resolution of singularities algorithm of [G4] to provide new estimates for exponential sums as well as new bounds on how often a function f(x) such as a polynomial with integer coefficients is divisible by various powers of a…

In his paper, W. M. Schmidt obtained an exponential sum estimate for systems of polynomials not including linear polynomials, which was then used to apply the Hardy-Littlewood circle method. We prove an analogous estimate for systems…

In this paper we consider the Newton polygons of $L$-functions coming from additive exponential sums associated to a polynomial over a finite field $\F_q$. These polygons define a stratification of the space of polynomials of fixed degree.…

Gelfond and Khovanskii found a formula for the sum of the values of a Laurent polynomial at the zeros of a system of n Laurent polynomials in the complex n-torus whose Newton polyhedra have generic mutual positions. An exponential change of…

In this paper we construct a Dwork theory for general exponential sums over affinoids in Witt towers. Using this, we compute the degree of the $L$-function, its Hodge polygon and examine when the Hodge and Newton polygons coincide.

We consider families of exponential sums indexed by a subgroup of invertible classes modulo some prime power $q$. For fixed $d$, we restrict to moduli $q$ so that there is a unique subgroup of invertible classes modulo $q$ of order $d$. We…

We prove some improvements of the classical Weil bound for one variable additive and multiplicative character sums associated to a polynomial over a finite field $k=\Fq$ for two classes of polynomials which are invariant under a large…

Let $R$ be a commutative ring, $f \in R[X_1,\ldots,X_k]$ a multivariate polynomial, and $G$ a finite subgroup of the group of units of $R$ satisfying a certain constraint, which always holds if $R$ is a field. Then, we evaluate $\sum…

We adapt ideas of Phong, Stein and Sturm and ideas of Ikromov and M\"uller from the continuous setting to various discrete settings, obtaining sharp bounds for exponential sums and the number of solutions to polynomial congruences for…

In this paper we generalize a result of Montgomery and Vaughan regarding exponential sums with multiplicative coefcients to the setting of Weyl sums. As applications, we establish a joint equidistribution result for roots of polynomial…