On Katz’s (A,B)-Exponential Sums

Abstract

Abstract We deduce Katz’s theorems for (A, B)-exponential sums over finite fields using $\ell$-adic cohomology and a theorem of Denef–Loeser, removing the hypothesis that A + B is relatively prime to the characteristic p. In some degenerate cases, the Betti number estimate is improved using toric decomposition and Adolphson–Sperber’s bounds for degrees of L-functions. Applying the facial decomposition theorem, we prove that the universal family of (A, B)-polynomials is generically ordinary for its L-function when p is in certain arithmetic progression.