We obtain general estimates for exponential integrals of the form \[ E_f(y)=\int_{\mathbb{Z}_{p}^{n}}ψ(\sum_{j=1}^r y_j f_j(x))|dx|, \] where the $f_j$ are restricted power series over $\mathbb{Q}_p$, $y_j\in\mathbb{Q}_p$, and $ψ$ a nontrivial additive character on $\mathbb{Q}_p$. We prove that if $(f_1,...,f_r)$ is a dominant map, then $|E_f(y)| < c|y|^α$ for some $c>0$ and $α<0$, uniform in $y$, where $|y|=\max(|y_i|)_i$. In fact, we obtain similar estimates for a much bigger class of exponential integrals. To prove these estimates we introduce a new method to study exponential sums, namely, we use the theory of $p$-adic subanalytic sets and $p$-adic integration techniques based on $p$-adic cell decomposition. We compare our results to some elementarily obtained explicit bounds for $E_f$ with $f_j$ polynomials.
This thesis contains some new results on the uniform rationality of Poincaré series of p-adic equivalence relations and Igusa's conjecture on exponential sums
Igusa and Denef-Sperber Conjectures on nondegenerate p-adic exponential sums
我们证明了[Igusa，J.，“高等学位形式讲座”，Lect. math. phys.，Springer-Verlag，59（1978）]和[Denef，J.和Sperber，S.，“Exponential sums mod p^n and Newton polyhedra”，Bull. Belg.数学会，Suppl.（2001）55-63]关于模p^m的非退化指数和。
Exponential sums: questions by Denef, Sperber, and Igusa
我们证明了Denef和Sperber猜想的其余部分[Denef，J.和Sperber，S.，\textit{Exponential sums mod $p^n$ and {N}ewton polyhedra}，布尔贝尔格数学会，{ \bf{suppl.} }（2001）55-63]关于模$p^m$的非退化局部指数和。我们推广了Igusa的猜想[Igusa，J.，\textit{Lectures on forms of higher degree}，Lect. math. phys.，Springer-Verlag，{ \bf{59} }（1978）]从齐次到准齐次的情形，证明了非退化情形和模$p$情形。本文推广了Katz [Katz，N. M.，\textit{Estimates for "singular" exponential sums}，Internat. Math. Res. Notices（1999）no.16，875-899]关于准齐次情形的有限域指数和。
We prove an upper bound on the log canonical threshold of a hypersurface that satisfies a certain power condition and use it to prove several generalizations of Igusa's conjecture on exponential sums, with the log-canonical threshold in the exponent of the estimates. We show that this covers optimally all situations of the conjectures for non-rational singularities, by comparing the log canonical threshold with a local notion of the motivic oscillation index.
We study the twisted local zeta function associated to a polynomial in two variables with coefficients in a non-Archimedean local field of arbitrary characteristic. Under the hypothesis that the polynomial is arithmetically non degenerate, we obtain an explicit list of candidates for the poles in terms of geometric data obtained from a family of arithmetic Newton polygons attached to the polynomial. The notion of arithmetical non degeneracy due to Saia and Zúñiga-Galindo is weaker than the usual notion of non degeneracy due to Kouchnirenko. As an application we obtain asymptotic expansions for…
Let $f$ be a polynomial in $n$ variables over some number field and $Z$ a subscheme of affine $n$-space. The notion of motivic oscillation index of $f$ at $Z$ was initiated by Cluckers (2008) and Cluckers-Mustaţǎ-Nguyen (2019). In this paper we elaborate on this notion and raise several questions. The first one is stability under base field extension; this question is linked to a deep understanding of the density of non-archimedean local fields over which Igusa's local zeta functions of $f$ has a pole with given real part. The second one is around Igusa's conjecture for exponential sums with b…
In this paper, we will give two proofs of the Cluckers-Veys conjecture on exponential sums for the case of polynomials in $\mathbb{Z}[x_{1},\ldots,x_{n}]$ having log-canonical thresholds at most one half. In particular, these results imply Igusa's conjecture and Denef-Sperber's conjecture under the same restriction on the log-canonical threshold.
We prove a recent conjecture due to Cluckers and Veys on exponential sums modulo $p^m$ for $m \geq 2$ in the special case where the phase polynomial $f$ is sufficiently non-degenerate with respect to its Newton polyhedron at the origin. Our main auxiliary result is an improved bound on certain related exponential sums over finite fields. This bound can also be used to settle a conjecture of Denef and Hoornaert on the candidate-leading Taylor coefficient of Igusa's local zeta function associated to a non-degenerate polynomial, at its largest non-trivial real candidate pole.
Let $\mathbf{F}_q$ be a finite field of $q$ elements. We show that the normalized Jacobi sum $q^{-(m-1)/2}J(χ_1,\dots,χ_m)$ ($χ_1\dotsm χ_m$ nontrivial) is asymptotically equidistributed on the unit circle, when $χ_1\in \mathcal{A}_1,\dots, χ_m\in \mathcal{A}_m$ run through arbitrary sets of nontrivial multiplicative characters of $\mathbf{F}_q^\times$, if $\#\mathcal{A}_1\ge q^{\frac{1}{2}+ε}$, $\#\mathcal{A}_2 \ge (\log q)^{\frac{1}δ-1}$ for $ε>δ>0$ fixed and $q\to \infty$ or if $\#\mathcal{A}_1\#\mathcal{A}_2/q\to \infty$. This extends previous results of Xi, Z. Zheng, and the authors.
On a uniform bound for exponential sums modulo $p^m$ for Deligne polynomials
KU Leuven（荷语鲁汶大学）；Institute of Mathematics, Vietnam Academy of Science and Technology（越南科学技术研究院数学研究所）；Normandie Université, Université de Caen Normandie - CNRS, Laboratoire de Mathématiques Nicolas Oresme (LMNO)（诺曼底大学，卡昂诺曼底大学-法国国家科学研究中心，尼古拉·奥雷数学实验室）
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 $\sum f(x_1,\ldots,x_k)$, where the summation is taken over all pairwise distinct $x_1,\ldots,x_k \in G$. In particular, let $p^s$ be a power of an odd prime, $n$ a positive integer coprime with $p-1$, and $a_1,\ldots,a_k$ integers such that $φ(p^s)$ divides $a_1+\cdots+a_k$ and $p-1$ does not divide $\sum_{i \in I}a_i$ for all non-empty proper subsets $I\subseteq \{1,\…
This paper proves Burgess bounds for short mixed character sums in multi-dimensional settings. The mixed character sums we consider involve both an exponential evaluated at a real-valued multivariate polynomial, and a product of multiplicative Dirichlet characters. We combine a multi-dimensional Burgess method with recent results on multi-dimensional Vinogradov Mean Value Theorems for translation-dilation invariant systems in order to prove character sum bounds in any dimension that recapture the Burgess bound in dimension 1. Moreover, we show that by embedding the given polynomial into an adv…
We obtain a nontrivial bound on the number of solutions to the equation $A^{x_1} + A^{x_2} = A^{x_3} + A^{x_4}$, $1 \le x_1,x_2,x_3,x_4 \le τ$, with a fixed $n\times n$ matrix $A$ over a finite field ${\mathbb F}_q$ of $q$ elements of multiplicative order $τ$. For $n=2$ this equation has been considered by Kurlberg and Rudnick (2001) in their study of quantum ergodicity for linear maps over ${\mathbb F}_q$. Furthermore, its multivariate analogue (also with $n=2$) has been studied by Bourgain (2005). We give applications of our result to obtaining a new bound of additive character sums with a m…
