id: 05712146
dt: j
an: 05712146
au: Gillot, Valérie; Langevin, Philippe
ti: Estimation of some exponential sum by means of $q$-degree.
so: Glasg. Math. J. 52, No. 2, 315-324 (2010).
py: 2010
pu: Cambridge University Press, Cambridge
la: EN
cc: 
ut: finite fields; Fourier coefficient; spectral amplitude
ci: Zbl 0834.11055; Zbl 0731.94010
li: doi:10.1017/S0017089510000017
ab: Let $K$ be the finite field of order $q$, a power of a prime $p$, and let
    $L$ be an extension of $K$ of degree $m$. The Fourier coefficient of
    $f(x)\in L[x]$ at $a\in L$ is: $$ \hat{f}(a)=\sum_{x\in L}
    μ_L(f(x)+ax), $$ where $μ_L$ is the canonical additive character of
    $L$. The spectral amplitude of $f$ is $R_L(f)=\max_{a\in L}
    |\hat{f}(a)|$. For a positive integer $d$, the $q$-weight is
    $wt_q(d)=\sum d_i$ where $d=\sum d_iq^i$. The main result here is that:
    $$ \max_{b\in L^*} R_L(bx^d)\le (wt_q(d)-1)^mq^{m/2}. $$ For
    comparison, the Carlitz-Uchiyama bound yields $\max_{b\in L^*}
    R_L(bx^d)\le (d-1)q^{m/2}$ when $p$ does not divide $d$. The technique
    is to pick a basis $β_i$ for $L$ over $K$ and apply a bound of
    Deligne’s to $F(x_1, \ldots ,x_m)=Tr_{L/K}(f(\sum x_iβ_i))$.
rv: Robert Fitzgerald (Carbondale)