Newton polyhedra and the degree of the L-function associated to an exponential sum
Let k be a finite field, \(k_ m\) its extension of degree m and \(\Psi\) a nontrivial additive character of k. The authors consider the L-function \(L^*(f,t)\) attached to a Laurent polynomial \(f\in k[x_ 1,x_ 1^{- 1},...,x_ n,x_ n^{-1}]\) via the exponential sums \(S^*_ m(f)=\sum \Psi \circ Tr_ m(f(x_ 1,...,x_ n))\) by \(L^*(f,t)=\exp (\sum^{\infty}_{m=1}S^*_ m(f)t^ m/m),\) where the sum runs over all \((x_ 1,...,x_ n)\in (k_ m^{\times})^ n\). These L-functions are always rational functions by work of Dwork and Grothendieck. Let \(V(f)\) be the volume of the Newton polyhedron \(\Delta(f) \subseteq {\mathbb{R}}^ n\) with respect to Lebesgue measure on \({\mathbb{R}}^ n\). Let \(\tilde n\) be the dimension of \(\Delta(f)\) and \(\tilde V(f)\) the \(\tilde n\)-dimensional volume of \(\Delta\) (f) relative to the induced lattice on the \(\tilde n\)-dimensional subspace containing \(\Delta\) (f). The main result is that for \(\tilde n\geq 1\) the following inequalities hold: \[ 0 \leq \text{degree}(L^*(f,t)^{(-1)^{n-1}}) \leq n! V(f), \] \[ \text{total degree}(L^*(f,t)) \leq 2^{\tilde n}(1+2^{(\tilde n+1)/\tilde n})^ n \tilde n! \tilde V(f), \] in particular for \(\tilde n<n\) one has degree\((L^*(f,t))=0\). This strengthens results obtained by \textit{E. Bombieri} [Invent. Math. 47, 29-39 (1978; Zbl 0396.14001)]. The first estimate is sharp, namely if f is nondegenerate in a certain sense then the upper bound is achieved. As a by-product the method delivers p- adic information about the roots of \(L^*(f,t)^{(-1)^{n-1}}\) in case this is a polynomial, thus generalizing a theorem of \textit{E. Bombieri} [Am. J. Math. 88, 71-105 (1966; Zbl 0171.415)].
scientific article; zbMATH DE number 4010640
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