# $p$-adic Integral Geometry

## TL;DR

This paper establishes a $p$-adic integral geometry formula for algebraic sets, providing bounds on intersection points and insights into random $p$-adic polynomial systems, extending classical geometric results into the $p$-adic setting.

## Contribution

It introduces a $p$-adic version of the integral geometry formula and applies it to intersection bounds and the analysis of random polynomial systems.

## Key findings

- Proves a $p$-adic integral geometry formula for algebraic sets.
- Provides bounds on the number of points in modulo $p^m$ reductions.
- Analyzes properties of random $p$-adic polynomial systems.

## 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 Oesterl\'e) and to the study of random $p$-adic polynomial systems of equations.