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

This graduate textbook provides an approach to estimating the isolated solutions of polynomial equations in n variables over an algebraically closed field through toric geometry. It collects and synthesizes works on Bernsteins theorem of counting solutions,extending it to affine space and applying it to derive Kushnirenko's results on Milnor numbers. The book aims to present the material in an elementary format, suitable for second-year graduate students.

Format: Hardback
Length: 352 pages
Publication date: 07 November 2021
Publisher: Springer Nature Switzerland AG

This comprehensive graduate textbook delves into the realm of toric geometry, offering a novel approach to estimating the isolated solutions (counted with appropriate multiplicity) of n polynomial equations in n variables over an algebraically closed field. Through the lens of toric geometry, the text assembles and synthesizes a diverse range of works on Bernsteins theorem of counting solutions of generic systems, culminating in a comprehensive presentation of the theorem, commentary, and extensions. The journey begins with Bernsteins original theorem, which elucidates solutions of generic systems in terms of the mixed volume of their Newton polytopes, encompassing comprehensive proofs of its recent extension to affine space and applications to open problems. Moreover, the text employs the developed techniques to derive and generalize Kushnirenko's remarkable results on Milnor numbers of hypersurface singularities, serving as a foundational precursor to the development of toric geometry. Ultimately, the book endeavors to present the material in an accessible and elementary format, nurturing the development of all requisite algebraic geometry to provide a thorough and accessible overview suitable for second-year graduate students.

Weight: 742g
Dimension: 163 x 241 x 25 (mm)
ISBN-13: 9783030751739
Edition number: 1st ed. 2021