How Many Zeroes?
Counting Solutions of Systems of Polynomials via Toric Geometry at Infinity
(Sprache: Englisch)
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...
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Klappentext zu „How Many Zeroes? “
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.
Inhaltsverzeichnis zu „How Many Zeroes? “
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.
Autoren-Porträt von Pinaki Mondal
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.
Bibliographische Angaben
- Autor: Pinaki Mondal
- 2022, 1st ed. 2021, XV, 352 Seiten, 81 farbige Abbildungen, Masse: 15,6 x 24 cm, Kartoniert (TB), Englisch
- Verlag: Springer, Berlin
- ISBN-10: 3030751767
- ISBN-13: 9783030751760
Sprache:
Englisch
Pressezitat
"The book will appeal to a reader interested on the arithmetic aspects of some natural intersections and interactions between algebraic and convex geometry." (Felipe Zaldívar, zbMATH 1483.13001, 2022)
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