ChevalleyWarning

The Chevalley–Warning theorem, sometimes referred to simply as Chevalley–Warning, is a result in finite field theory and algebraic geometry that concerns the number of common solutions to polynomial equations over finite fields. It provides a divisibility property for the number of solutions under a constraint on the total degrees of the polynomials.

Let F be a finite field with q elements, where q = p^r and p is its characteristic. Consider polynomials f1, ..., fm in F[x1, ..., xn] with total degrees d1, ..., dm, and set d = d1 + ... + dm. If d < n, then the number N of vectors a in F^n that satisfy f1(a) = ... = fm(a) = 0 is divisible by p. In particular, if the system has any solution, it has at least p distinct solutions. For homogeneous systems, where every polynomial has zero constant term, this implication guarantees the existence of a nontrivial (nonzero) solution.

The theorem emerged from work by Claude Chevalley and Émile Warning in the mid-1930s and is named in their honor. It has several generalizations and related results, including extensions to broader classes of polynomials, various refinements of the divisibility statements, and adaptations to different algebraic contexts. The Chevalley–Warning theorem is widely cited for its elementary yet powerful consequence in counting and existence arguments for polynomial equations over finite fields.

Applications of the result appear in number theory, combinatorics, and the study of polynomial mappings over finite fields, where the divisibility property yields nontrivial information about the structure and existence of solutions.