How Many Zeroes? Counting Solutions of Systems of Polynomials via Toric Geometry at Infinity

Mathematics Subject Classification: • 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry • 14M25 Toric varieties, Newton polyhedra, Okounkov bodies • 52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) • 14N10 Enumerative problems (combinatorial problems) in algebraic geometry

This graduate textbook presents an approach through toric geometry to the problem of estimating the isolated solutions (counted with appropriate multiplicity) of n polynomial equations in n variables over an algebraically closed field. The text collects and synthesizes a number of works on Bernstein’s theorem of counting solutions of generic systems, ultimately presenting the theorem, commentary, and extensions in a comprehensive and coherent manner. It begins with Bernstein’s original theorem expressing solutions of generic systems in terms of the mixed volume of their Newton polytopes, including complete proofs of its recent extension to affine space and some applications to open problems. The text also applies the developed techniques to derive and generalize Kushnirenko's results on Milnor numbers of hypersurface singularities, which has served as a precursor to the development of toric geometry. Ultimately, the book aims to present material in an elementary format, developing all necessary algebraic geometry to provide a truly accessible overview suitable to second-year graduate students.