$p$-adic Integral Geometry

Abstract

We prove a $p$-adic version of the Integral Geometry Formula for averaging the intersection of two $p$-adic projective algebraic sets. We apply this result to give bounds on the number of points in the modulo $p^m$ reduction of a projective set (reproving a result by Oesterlé) and to the study of random $p$-adic polynomial systems of equations.