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.
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机译：对于n变量中f1，...，fr的齐次多项式，其系数在Fq中，并且在Fq，chi1，...，chir上具有乘性，我们检查总和Sm =x∈＆parl0; P n-1Fqm c 1f1x ... crf rx其中Sm是通过将字符chij从Fq扩展到Fqm来定义的。特别地，我们证明了与这些和Lt = exp m = 1infinitySm tm / m关联的L函数是一个多项式，并且根据q，r和fj的度数找到了一个公式。所有计算均使用Dwork的同调理论完成。在多项式f1，...，fr定义一个具有正交的除数的假设下，除一维外，所有同调性都将消失。我们还使用作用于p-adic幂级数的Banach空间上的Frobenius算子矩阵，计算L函数的p-adic牛顿多边形的下界。
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