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

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Pinaki Mondal

How Many Zeroes?

Counting Solutions of Systems of Polynomials via Toric Geometry at Infinity

Pinaki Mondal

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English

Springer Nature Switzerland AG
07 November 2022

Algebraic geometry

Series: CMS/CAIMS Books in Mathematics

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.

By: Pinaki Mondal
Imprint: Springer Nature Switzerland AG
Country of Publication: Switzerland
Edition: 1st ed. 2021
Volume: 2
Dimensions: Height: 235mm, Width: 155mm,
Weight: 563g
ISBN: 9783030751760
ISBN 10: 3030751767
Series: CMS/CAIMS Books in Mathematics
Pages: 352
Publication Date: 07 November 2022
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

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) Bézout 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.

Pinaki Mondal studied at Khulna St. Joseph's School, Barisal Cadet College, University of Saskatchewan and University of Toronto. After a postdoctoral fellowship at the Weizmann Institute and teaching at the University of The Bahamas, he is back in Toronto doing quantitative finance. When not working to safeguard Canadian economy from a collapse, he still makes time to think about algebraic geometry.

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

The book will appeal to a reader interested on the arithmetic aspects of some natural intersections and interactions between algebraic and convex geometry. (Felipe Zaldivar, zbMATH 1483.13001, 2022)