How many zeroes? Counting the number of solutions of systems of polynomials via geometry at infinity (Draft III)

TL;DR

This paper presents a comprehensive approach using toric geometry to estimate the number of solutions of polynomial systems over algebraically closed fields, extending classical results from the torus to affine space.

Contribution

It provides a complete proof of Bernstein's theorem and its recent extension to affine space, along with applications to Milnor numbers of hypersurface singularities.

Findings

Extended Bernstein's theorem to affine space

Derived explicit formulas for solution counts

Connected solution counting to Newton polytopes and mixed volumes

Abstract

In this book we describe an approach through toric geometry to the following problem: "estimate the number (counted with appropriate multiplicity) of isolated solutions of n polynomial equations in n variables over an algebraically closed field k." The outcome of this approach is the number of solutions for "generic" systems in terms of their "Newton polytopes," and an explicit characterization of what makes a system "generic." The pioneering work in this field was done in the 1970s by Kushnirenko, Bernstein and Khovanskii, who completely solved the problem of counting solutions of generic systems on the "torus" (k\0)^n. In the context of our problem, however, the natural domain of solutions is not the torus, but the affine space k^n. There were a number of works on extension of Bernstein's theorem to the case of affine space, and recently it has been completely resolved, the final steps having been carried out by the author. The aim of this book is to present these results in a coherent way. We start from the beginning, namely Bernstein's beautiful theorem which expresses the number of solutions of generic systems in terms of the mixed volume of their Newton polytopes. We give complete proofs, over arbitrary algebraically closed fields, of Bernstein's theorem and its recent extension to the affine space, and describe some open problems. We also apply the developed techniques to derive and generalize Kushnirenko's results on Milnor numbers of hypersurface singularities which in 1970s served as a precursor to the development of toric geometry. Care was taken to make this book as elementary as possible. In particular, it develops all the necessary algebraic geometry (modulo some explicitly stated basic results) with lots of examples and exercises, and can be used as a quick introduction to basic algebraic geometry.