Abstract :
Let f(x) be a polynomial with p-adic coefficients, and defineimage where χ is a character of (image/pmimage)×. For example, f(x)=x leads to the Gauss sum of χ. An explicit formula for S(f;χ;image:/pmimage) is given if f′(x) has distinct roots in image/pmimage and mgreater-or-equal, slanted2, or if f′(x) has distinct roots p-adically and m is sufficiently large. In either case, the formula involves one term for each root of f′(x). If the derivative does not have distinct roots, there is a less explicit formula, requiring one term for each approximate root. Analogous results are proved for sums involving several variables. As applications of these results, Mauclaireʹs evaluation of the Gauss sum is re-derived, and new results on n-variable Kloosterman sums are obtained.
