The Lefschetz Fixed Point Theorem and Solutions to Polynomials over Finite Fields

Suppose we have an equation with integer coefficients, e.g. y = x + x and we want to understand its solutions over a finite field. If we consider the solutions over C, the set of solutions is a complex manifold and can be studied using the powerful tools of complex analysis and algebraic topology. However, since the set of solutions over a finite field is also finite, and any obvious topology makes the set into a finite discrete set, there is little hope of using topological tools over the finite field. The goal of this paper is to demonstrate the idea that counting points provides a good replacement for algebraic topology over finite fields by analyzing some examples. We begin by counting points in some specific examples of algebraic varieties, which are sets that can be locally described as the zero set of some polynomial.