Related papers: On a uniform bound for exponential sums modulo $p^…

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 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…

We prove the remaining part of the conjecture by Denef and Sperber [Denef, J. and Sperber, S., \textit{Exponential sums mod $p^n$ and {N}ewton polyhedra}, Bull. Belg. Math. Soc., {\bf{suppl.}} (2001) 55-63] on nondegenerate local…

We combine two of Igusa's conjectures with recent semi-continuity results by Musta\c{t}\u{a} and Popa to form a new, natural conjecture about bounds for exponential sums. These bounds have a deceivingly simple and general formulation in…

We propose a conjecture for exponential sums which generalizes both a conjecture by Igusa and a local variant by Denef and Sperber, in particular, it is without the homogeneity condition on the polynomial in the phase, and with new…

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-Musta\c{t}\v{a}-Nguyen (2019). In…

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…

We prove the intersection of Igusa's Conjecture of [Igusa, J., "Lectures on forms of higher degree", Lect. math. phys., Springer-Verlag, 59 (1978)] and the Denef - Sperber Conjecture of [Denef, J. and Sperber, S., "Exponential sums mod p^n…

We consider incomplete exponential sums in several variables of the form S(f,n,m) = \frac{1}{2^n} \sum_{x_1 \in \{-1,1\}} ... \sum_{x_n \in \{-1,1\}} x_1 ... x_n e^{2\pi i f(x)/p}, where m>1 is odd and f is a polynomial of degree d with…

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…

We obtain general estimates for exponential integrals of the form \[ E_f(y)=\int_{\mathbb{Z}_{p}^{n}}\psi(\sum_{j=1}^r y_j f_j(x))|dx|, \] where the $f_j$ are restricted power series over $\mathbb{Q}_p$, $y_j\in\mathbb{Q}_p$, and $\psi$ a…

This thesis contains some new results on the uniform rationality of Poincar\'e series of p-adic equivalence relations and Igusa's conjecture on exponential sums

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…

In 2006, Budur, Musta\c{t}\v{a} and Saito introduced the notion of Bernstein-Sato polynomial of an arbitrary scheme of finite type over fields of characteristic zero. Because of the strong monodromy conjecture, it should have a…

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…

In this note we consider algebraic exponential sums over the values of homogeneous nonsingular polynomials $F(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$ in the quotient ring $\mathbb{Z}/p^2\mathbb{Z}$. We provide an estimate of…

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,…

This is an expository paper aiming to introduce Zilber's Exponential Closedness conjecture to a general audience. Exponential Closedness predicts when (systems of) equations involving addition, multiplication, and exponentiation have…

Let p be a prime and let F_pbar be the algebraic closure of the finite field of p elements. Let f(x) be any one variable rational function over F_pbar with n poles of orders d_1, ...,d_n. Suppose p is coprime to d_i for every i. We prove…

We obtain asymptotic expansions for local probabilities of partial sums for uniformly bounded independent but not necessarily identically distributed integer-valued random variables. The expansions involve products of polynomials and…