Finite Field Experiments (with an Appendix by Stefan Wiedmann)

TL;DR

This paper discusses using computer experiments over finite fields to heuristically analyze solutions of polynomial equations in characteristic zero, providing a tutorial overview of methods and applications.

Contribution

It introduces a methodology for leveraging finite field experiments to gain heuristic insights into polynomial solution sets in characteristic zero, with practical tutorial guidance.

Findings

Finite field experiments can provide useful heuristic information.

The paper offers a tutorial approach for applying these experiments.

Applications include understanding solution structures of polynomial equations.

Abstract

We explain how to use computer experiments over finite fields to gain heuristic information about the solution set of polynomial equations in characteristic zero. These are notes of a tutorial I gave at the NATO Advanced Study Institute on Higher-Dimensional Geometry over Finite Fields in G"ottingen 2007.