| For f1,...,fr homogeneous polynomials in n variables with coefficients in Fq , and multiplicative characters on Fq , chi1,...,chir we examine the sum Sm=x∈&parl0;P n-1Fqm c 1f1x ...crf rx where Sm is defined by extending the characters chij from Fq to Fqm . In particular we show that the L-function associated to these sums Lt=exp m=1infinitySm tm/m, is a polynomial and we find a formula for its degree in terms of q, r and the degrees of the fj. All computations are done using Dwork's cohomology theory. Under the hypothesis that the polynomials f1,...,fr define a divisor with normal crossings, then the cohomology will vanish for all but one dimension. We also compute a lower bound for the p-adic Newton polygon of the L-function using the matrix of the Frobenius operator acting on a certain Banach space of p-adic power series. |
