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Search: MSC category 11L05 ( Gauss and Kloosterman sums; generalizations )

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1. CMB 2007 (vol 50 pp. 71)

Gurak, S.
Polynomials for Kloosterman Sums
Fix an integer $m>1$, and set $\zeta_{m}=\exp(2\pi i/m)$. Let ${\bar x}$ denote the multiplicative inverse of $x$ modulo $m$. The Kloosterman sums $R(d)=\sum_{x} \zeta_{m}^{x + d{\bar x}}$, $1 \leq d \leq m$, $(d,m)=1$, satisfy the polynomial $$f_{m}(x) = \prod_{d} (x-R(d)) = x^{\phi(m)} +c_{1} x^{\phi(m)-1} + \dots + c_{\phi(m)},$$ where the sum and product are taken over a complete system of reduced residues modulo $m$. Here we give a natural factorization of $f_{m}(x)$, namely, $$ f_{m}(x) = \prod_{\sigma} f_{m}^{(\sigma)}(x),$$ where $\sigma$ runs through the square classes of the group ${\bf Z}_{m}^{*}$ of reduced residues modulo $m$. Questions concerning the explicit determination of the factors $f_{m}^{(\sigma)}(x)$ (or at least their beginning coefficients), their reducibility over the rational field ${\bf Q}$ and duplication among the factors are studied. The treatment is similar to what has been done for period polynomials for finite fields.

Categories:11L05, 11T24

2. CMB 2003 (vol 46 pp. 344)

Gurak, S.
Gauss and Eisenstein Sums of Order Twelve
Let $q=p^{r}$ with $p$ an odd prime, and $\mathbf{F}_{q}$ denote the finite field of $q$ elements. Let $\Tr\colon\mathbf{F}_{q} \to\mathbf{F}_{p} $ be the usual trace map and set $\zeta_{p} =\exp(2\pi i/p)$. For any positive integer $e$, define the (modified) Gauss sum $g_{r}(e)$ by $$ g_{r}(e) =\sum_{x\in \mathbf{F}_{q}}\zeta_{p}^{\Tr x^{e}} $$ Recently, Evans gave an elegant determination of $g_{1}(12)$ in terms of $g_{1}(3)$, $g_{1}(4)$ and $g_{1}(6)$ which resolved a sign ambiguity present in a previous evaluation. Here I generalize Evans' result to give a complete determination of the sum $g_{r}(12)$.

Categories:11L05, 11T24

3. CMB 2001 (vol 44 pp. 22)

Evans, Ronald
Gauss Sums of Orders Six and Twelve
Precise, elegant evaluations are given for Gauss sums of orders six and twelve.

Categories:11L05, 11T24

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