Overview
- First textbook containing complete proofs of various weighted versions of Bézout's theorem, Bernstein's theorem and its extension to the affine space
- Gives a new proof of, and generalizes, Kushnirenko's results on Milnor number of non-degenerate singularities
- Develops necessary algebraic geometry and convex geometry prerequisites quickly with the help of numerous exercises
Part of the book series: CMS/CAIMS Books in Mathematics (CMS/CAIMS BM, volume 2)
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Table of contents (12 chapters)
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Front Matter
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Preliminaries
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Front Matter
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Number of Zeroes on the Torus
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Front Matter
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Back Matter
Keywords
About this book
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.
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Bibliographic Information
Book Title: How Many Zeroes?
Book Subtitle: Counting Solutions of Systems of Polynomials via Toric Geometry at Infinity
Authors: Pinaki Mondal
Series Title: CMS/CAIMS Books in Mathematics
DOI: https://doi.org/10.1007/978-3-030-75174-6
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
Hardcover ISBN: 978-3-030-75173-9Published: 07 November 2021
Softcover ISBN: 978-3-030-75176-0Published: 07 November 2022
eBook ISBN: 978-3-030-75174-6Published: 07 November 2021
Series ISSN: 2730-650X
Series E-ISSN: 2730-6518
Edition Number: 1
Number of Pages: XV, 352
Number of Illustrations: 7 b/w illustrations, 81 illustrations in colour
Topics: Algebraic Geometry