How Many Zeroes?: Counting Solutions of Systems of Polynomials via Toric Geometry at Infinity [Paperback]

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 Bernsteins theorem of counting solutions of generic systems, ultimately presenting the theorem, commentary, and extensions in a comprehensive and coherent manner. It begins with Bernsteins 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.

Introduction.- A brief history of points of infinity in geometry.- Quasiprojective varieties over algebraically closed fields.- Intersection multiplicity.- Convex polyhedra.- Toric varieties over algebraically closed fields.- Number of solutions on the torus: BKK bound.- Number of zeroes on the affine space I: (Weighted) B?zout theorems.- Intersection multiplicity at the origin.- Number of zeroes on the affine space II: the general case.- Minor number of a hypersurface at the origin.- Beyond this book.- Miscellaneous commutative algebra.- Some results related to schemes.- Notation.- Bibliography.The book will appeal to a reader interested on the arithmetic aspects of some natural intersections and interactions between algebraic and convex geometry. (Felipe Zald?var, zbMATH 1483.13001, 2022)Pinaki Mondal studied at Khulna St. Joseph's School, Barisal Cadet lã0