Pinaki Mondal: How Many Zeroes? | Taschenbuch

Cover: 9783030751760 | How Many Zeroes? | Pinaki Mondal | Taschenbuch | xv | Englisch | 2022

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Sprache: Englisch

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Beschreibung

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.

Über den Autor

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.

Zusammenfassung

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

Inhaltsverzeichnis

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.

Details

Erscheinungsjahr:	2022
Fachbereich:	Arithmetik & Algebra
Genre:	Mathematik , Medizin , Naturwissenschaften , Technik
Rubrik:	Naturwissenschaften & Technik
Medium:	Taschenbuch
Inhalt:	xv 352 S. 7 s/w Illustr. 81 farbige Illustr. 352 p. 88 illus. 81 illus. in color.
ISBN-13:	9783030751760
ISBN-10:	3030751767
Sprache:	Englisch
Einband:	Kartoniert / Broschiert
Autor:	Mondal, Pinaki
Auflage:	1st edition 2021
Hersteller:	Springer International Publishing
Verantwortliche Person für die EU:	Springer Verlag GmbH, Tiergartenstr. 17, D-69121 Heidelberg, juergen.hartmann@springer.com
Maße:	235 x 155 x 20 mm
Von/Mit:	Pinaki Mondal
Erscheinungsdatum:	07.11.2022
Gewicht:	0,557 kg