Exponential sums: Questions by Denef, Sperber, and Igusa

Abstract:

We prove the remaining part of the conjecture by Denef and Sperber [Denef, J. and Sperber, S., Exponential sums mod $p^n$ and Newton polyhedra, Bull. Belg. Math. Soc., suppl. (2001) 55-63] on nondegenerate local exponential sums modulo $p^m$. We generalize Igusa’s conjecture in the introduction of [Igusa, J., Lectures on forms of higher degree, Lect. Math. Phys., Springer-Verlag, 59 (1978)] from the homogeneous to the quasi-homogeneous case and prove the nondegenerate case as well as the modulo $p$ case. We generalize some results by Katz in [Katz, N. M., Estimates for “singular” exponential sums, Internat. Math. Res. Notices (1999) no. 16, 875-899] on finite field exponential sums to the quasi-homogeneous case.