ChevalleyWarning, Beyond and Beneath Polynomial Equations Over Finite Rings

The Chevalley-Warning Theorem answers a fundamental question about a low degree polynomial system over a finite field: the number of solutions is divisible by the characteristic. Since its discovery in 1935, this result has become a powerful tool with applications across number theory, combinatorics, and theoretical computer science. This book explores the mathematics lying beyond and beneath the Chevalley-Warning Theorem. The first chapter goes beyond by treating stronger versions due to Ax-Katz (1964 and 1971) and Moreno-Moreno (1995), among others. By comparing multiple proofs of these classical results, the reader also learns what lies beneath them, uncovering a ""core mathematics"" that combines aspects of algebraic number theory, basic commutative algebra, combinatorics and interpolation theory but can be explored on its own terms. Further development of the mathematics underneath leads to generalizations of Chevalley-Warning (and beyond) to several other contexts: arbitrary finite rings, restricted variables and even to purely group-theoretic generalizations. The final chapter highlights applications of these ideas in additive combinatorics, discrete geometry, graph theory, and interpolation theory. With prerequisites limited to basic algebraic number theory, the book is particularly suitable for graduate students in number theory. At the same time, it will appeal to combinatorialists and to readers interested in modern developments of the polynomial method.