Explicit Evaluation of Certain Exponential Sums of Quadratic Functions over $\Bbb F_{p^n}$, $p$ Odd
| dc.creator | Draper, Sandra | |
| dc.creator | Hou, Xiang-dong | |
| dc.date | 2007-08-27 | |
| dc.date.accessioned | 2026-07-07T08:25:54Z | |
| dc.date.available | 2026-07-07T08:25:54Z | |
| dc.description | Let $p$ be an odd prime and let $f(x)=\sum_{i=1}^ka_ix^{p^{α_i}+1}\in\Bbb F_{p^n}[x]$, where $0\le α_1<...<α_k$. We consider the exponential sum $S(f,n)=\sum_{x\in\Bbb F_{p^n}}e_n(f(x))$, where $e_n(y)=e^{2πi\text{Tr}_n(y)/p}$, $y\in\Bbb F_{p^n}$, $\text{Tr}_n=\text{Tr}_{\Bbb F_{p^n}/\Bbb F_p}$. 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 $ν_p(m) \le \min\{ν_p(α_i):1\le i\le k\}$, where $ν_p$ is the $p$-adic order. We also find an explicit formula for $S(f,n)$ when $ν_2(α_1)=...= ν_2(α_k)<ν_2(n)$. 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. | |
| dc.description | 28 pages | |
| dc.identifier | https://arxiv.org/abs/0708.3619 | |
| dc.identifier | http://arxiv.org/abs/0708.3619 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/136772 | |
| dc.subject | Number Theory | |
| dc.subject | 11T23 | |
| dc.title | Explicit Evaluation of Certain Exponential Sums of Quadratic Functions over $\Bbb F_{p^n}$, $p$ Odd | |
| dc.type | text |