How Many Zeroes?
Counting Solutions of Systems of Polynomials via Toric Geometry at Infinity
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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 K. 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 a second-year graduate students.
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.
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
Autor*in
Pinaki Mondal
Themen in »How Many Zeroes?«
Number of solutions/zeros of systems of polynomials affine Bezout problem Bezout's theorem Bernstein-Kushnirenko theorem BKK theorem toric varieties intersection multiplicity Milnor number Newton number Non-degenerate polynomials
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“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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