Related papers: A sup-Hodge bound for exponential sums

We show that deciding whether a sparse univariate polynomial has a p-adic rational root can be done in NP for most inputs. We also prove a polynomial-time upper bound for trinomials with suitably generic p-adic Newton polygon. We thus…

In this paper, we obtain bounds on the $L^1$ norm of the sum $\sum_{n\le x}\tau(n) e(\alpha n)$ where $\tau(n)$ is the divisor function.

Let a planar algebraic curve $C$ be defined over a valuation field by an equation $F(x,y)=0$. Valuations of the coefficients of $F$ define a subdivision of the Newton polygon $\Delta$ of the curve $C$. If a given point $p$ is of…

Let $p$ be prime, $N$ be a positive integer prime to $p$, and $k$ be an integer. Let $P_k(t)$ be the characteristic series for Atkin's $U$ operator as an endomorphism of $p$-adic overconvergent modular forms of tame level $N$ and weight…

Let Delta be an integral convex polytope containing the origin of dimension n in the n-dim real space and it is simplicial at all origin-less facets. Let A(Delta) be the space of all Laurent polynomials f parametered by its coefficients…

In this paper, we mainly give a general explicit form of Cassels' $p$-adic embedding theorem for number fields. We also give its refined form in the case of cyclotomic fields. As a byproduct, given an irreducible polynomial $f$ over $Z$, we…

Let $p$ be an odd prime. What are the possible Newton polygons for a curve in characteristic $p$? Equivalently, which Newton strata intersect the Torelli locus in $\mathcal{A}_g$? In this note, we study the Newton polygons of certain curves…

Let $p$ be a prime number, $C$ be any absolutely irreducible affine plane curve over $\mathbb{F}_p$, and $g,f\in\mathbb{F}_p(x,y)$ be rational functions. We continue the study of the distribution of the values of short hybrid exponential…

The goal of this paper is to show a (derived) $p$-adic Simpson correspondence for (locally) unipotent coefficients on smooth rigid-analytic varieties. Our results depend on a deformation to $\mathbf{B}_\mathtt{dr}^+/\xi^2$, and not on a…

This paper proves a bound for exponential sums over orbits of vectors in $\mathbb{F}_p^d$ under subgroups of $\rm{GL}_d(\mathbb{F}_p)$. The main tool is a classification theorem for approximate groups due to Gill, Helfgott, Pyber, and…

In this paper we give a modular interpretation of the $k$-th symmetric power $L$-function of the Kloosterman family of exponential sums in characteristics 2 and 3, and in the case of $p=2$ and $k$ odd give the precise 2-adic Newton polygon.…

Power functions with low $c$-differential uniformity have been widely studied not only because of their strong resistance to multiplicative differential attacks, but also low implementation cost in hardware. Furthermore, the…

In the paper we prove a new upper bound for Heilbronn's exponential sum and obtain some applications of our result to distribution of Fermat quotients.

Let A be a finite set of integers and F_A its exponential sum. McGehee, Pigno & Smith and Konyagin have independently proved that the L^1-norm of F_A is at least c log|A| for some absolute constant c. The lower bound has the correct order…

In this paper we apply Ax-Schanuel's Theorem to the ultraproduct of $p$-adic fields in order to get some results towards algebraic independence of $p$-adic exponentials for almost all primes $p$.

We present lower bounds on the sum and product of the distinct prime factors of an odd perfect number, which provide a lower bound on the size of the odd perfect number as a function of the number of its distinct prime factors.

In this paper, we derive new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in absolute value at certain power are P-functions. Some applications to special means of real…

We investigate when the exponential sum $S_f(x,\alpha) := \sum_{n\le x}f(n)\mathrm{e}(n\alpha)$ is bounded, for a multiplicative function $f$ and $\alpha\in\mathbb{R}$. We show that under natural assumptions, $S_f(x,\alpha)$ is bounded only…

In this work we found order estimates for the upper bounds of the deviations of analogue of Zigmund's sums on the classes of $(\psi;\beta)$-differentiable functions in the metrics of generalized Lebesgue spaces with variable exponent.

We consider a q-analogue of the standard bilinear form on the commutative ring of symmetric functions. The q=-1 case leads to a Z-graded Hopf superalgebra which we call the algebra of odd symmetric functions. In the odd setting we describe…