Improvements of 𝑝-adic estimates of exponential sums

Feng, Yulu; Hong, Shaofang

doi:10.1090/proc/15995

Improvements of $p$-adic estimates of exponential sums
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by Yulu Feng and Shaofang Hong PDF

Proc. Amer. Math. Soc. 150 (2022), 3687-3698 Request permission

Abstract:

Let $n, r$ and $f$ be positive integers. Let $p$ be a prime number and $\psi$ be an arbitrary fixed nontrivial additive character of the finite field $\mathbb F_q$ with $q=p^f$ elements. Let $F$ be a polynomial in $\mathbb F_q[x_1,\dots ,x_n]$ and $V$ be the affine algebraic variety defined over $\mathbb {F}_q$ by the simultaneous vanishing of the polynomials $\{F_i\}_{i=1}^r\subseteq \mathbb F_q[x_1,\dots ,x_n]$. Let $\mathbb {Z}_{\ge 0}$ stand for the set of all nonnegative integers and $A$ be an arbitrary nonempty subset of $\{1,\dots ,n\}$. For a polynomial $H(X)=\sum _{{\mathbf {d}}}\alpha _{\mathbf {d}}X^{\mathbf {d}}$ with ${\mathbf {d}}=(d_1,\dots ,d_n)\in \mathbb {Z}_{\ge 0}^n, X^{\mathbf {d}}=x_1^{d_1}\dots x_n^{d_n}$ and $\alpha _{\mathbf {d}}\in \mathbb {F}_q^*$, we define $\deg _A(H)=\max _{{\mathbf {d}}}\{\sum _{i\in A}d_i\}$ to be the $A$-degree of $H$. In this paper, for the exponential sum $S(F,V,\psi )=\sum _{X\in V(\mathbb {F}_q)}\psi (F(X))$ with $V(\mathbb {F}_q)$ being the set of the $\mathbb {F}_q$-rational points of $V$, we show that \begin{equation*} \mathrm {ord}_q S(F,V,\psi )\ge \frac {|A|-\sum _{i=1}^r\deg _A(F_i)} {\max _{1\le i\le r}\{\deg _A(F),\deg _A(F_i)\}} \end{equation*} if $\deg _A(F)>0$ or $\deg _A(F_i)>0$ for some $i\in \{1,\dots ,r\}$. This estimate improves Sperber’s theorem obtained in 1986. This also leads to an improvement of the $p$-adic valuation of the number $N(V)$ of $\mathbb {F}_q$-rational points on the variety $V$ which strengthens the Ax-Katz theorem. Moreover, we use the $A$-degree and $p$-weight $A$-degree to establish $p$-adic estimates on multiplicative character sums and twisted exponential sums which improve Wan’s results gotten in 1995.

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