Exponential sums and polynomial congruences along p-adic submanifolds

In this article, we consider the estimation of exponential sums along the points of the reduction mod pm of a p-adic analytic submanifold of Zn p . More precisely, we extend Igusa’s stationary phase method to this type of exponential sums. We also study the number of solutions of a polynomial congruence along the points of the reduction mod pm of a p-adic analytic submanifold of Zn p . In addition, we attach a Poincaré series to these numbers, and establish its rationality. In this way, we obtain geometric bounds for the number of solutions of the corresponding polynomial congruences.

• Type: Journal
• Year: 2011
• Where: FFA
• Authors: Dirk Segers, W. A. Zuniga-Galindo
• Keywords: Exponential Sums, FFA 2011, Phase Method, Stationary Phase, VLSI