Explicit Evaluation of Certain Exponential Sums of Quadratic Functions over ${\mathbb{F}}_{p^n}$, $p$ Odd

Abstract

Let $p$ be an odd prime and let $f(x)={\sum }_{i=1}^k{a}_i{x}^{p^{{\alpha }_i}+1}\in {\mathbb{F}}_{p^n}[x]$, where $0\le {\alpha }_1<...<{\alpha }_k$. We consider the exponential sum $S(f,n)={\sum }_{x\in {\mathbb{F}}_{p^n}}{e}_n(f(x))$, where $e_n(y)=e^{2\pi i{\text{Tr}}_n(y)/p}$, $y\in {\mathbb{F}}_{p^n}$, ${\text{Tr}}_n={\text{Tr}}_{{\mathbb{F}}_{p^n}/{\mathbb{F}}_p}$. There is an effective way to compute the nullity of the quadratic form ${\text{Tr}}_{mn}(f(x))$ for all integer $m>0$. Assuming that all such nullities are known, we find relative formulas for $S(f,mn)$ in terms of $S(f,n)$ when ${\nu }_p(m)\le \min \{{\nu }_p({\alpha }_i):1\le i\le k\}$, where ${\nu }_p$ is the $p$-adic order. We also find an explicit formula for $S(f,n)$ when ${\nu }_2({\alpha }_1)=...={\nu }_2({\alpha }_k)<{\nu }_2(n)$. These results generalize those by Carlitz and by Baumert and McEliece. Parallel results with $p=2$ were obtained in a previous paper by the second author.

28 pages