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

Description

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. Preface Contents Chapter I. Introduction 1. The problem and the results 2. Prerequisites 3. Organization Part 1. Preliminaries Chapter III. Quasiprojective varieties over algebraically closed fields 1. Affine varieties 2. (Ir)reducibility 3. Regular functions, coordinate rings and morphisms of affine varieties 4. Quasiprojective varieties 5. Regular functions 6. Morphisms of quasiprojective varieties; affine varieties as quasiprojective varieties 7. Rational functions and rational maps on irreducible varieties 8. Product spaces, Segre map, Veronese embedding 9. Completeness and compactification 10. Image of a morphism: Part I 11. Dimension 12. Image of a morphism: Part II - Constructible sets 13. Tangent space, singularities, local ring at a point 14. Completion of the local ring at a point 15. Degree of a dominant morphism Chapter IV. *Intersection multiplicity 1. Introduction 2. Closed subschemes of a variety 3. Possibly non-reduced curves 4. Intersection multiplicity at a nonsingular point of a variety 5. Intersection multiplicity of complete intersections Chapter V. Convex polyhedra 1. Basic notions 2. Characterization of convex polyhedra 3. Basic properties of convex polyhedra 4. Normal fan of a convex polytope 5. Rational polyhedra 6. *Volume of convex polytopes 7. *Volume of special classes of polytopes Part 2. Number of zeroes on the torus Chapter VI. Toric varieties over algebraically closed fields 1. Algebraic torus 2. Toric varieties from finite subsets of mathbbZN 3. Examples of toric varieties 4. Structure of XmathcalA 5. Toric varieties from polytopes 6. Nonsingularity in codimension one on XmathcalP 7. Extending closed subschemes of the torus to XmathcalP 8. Branches of curves on the torus 9. Points at infinity on toric varieties 10. *Weighted projective spaces 11. *Weighted blow up Chapter VII. Number of zeroes on the torus: BKK bound 1. Introduction 2. Mixed volume 3. Theorems of Kushnirenko and Bernstein 4. Proof of Bernstein-Kushnirenko non-degeneracy condition 5. Proof of the BKK bound 6. Applications of Bernstein's theorem to convex geometry 7. Some technical results 8. The problem of characterizing coefficients which guarantee non-degeneracy 9. Notes Part 3. Beyond the torus Chapter VIII. Number of zeroes on the affine space I: (Weighted) Bézout theorems 1. Weighted degree 2. mathbbPn(ω) as a compactification of kn when the ωj are positive and ω0 = 1 3. Weighted Bézout theorem 4. Products of weighted projective spaces 5. Weighted multi-homogeneous Bézout theorem 6. Notes Chapter IX. Intersection multiplicity at the origin 1. Introduction 2. Generic intersection multiplicity 3. Characterization of minimal multiplicity systems 4. Proof of the non-degeneracy condition 5. Proof of the bound 6. The efficient version of the non-degeneracy condition 7. Other formulae for generic intersection multiplicity 8. Monotonicity of generic intersection multiplicity 9. Notes Chapter X. Number of zeroes on the affine space II: the general case 1. Introduction 2. The bound 3. Derivation of the formuale for the bound 4. Other formulae for the bound 5. Examples motivating the non-degeneracy conditions 6. Non-degeneracy conditions 7. Proof of the non-degeneracy conditions 8. Weighted Bézout theorem: general version 9. Weighted multi-homogeneous Bézout theorem: general version 10. Open problems Chapter XI. Milnor number of a hypersurface at the origin 1. Introduction 2. Milnor number 3. Generic Milnor number 4. Classical notions of non-degeneracy 5. Newton number: Kushnirenko's formula for the generic Milnor number 6. Open problems Chapter XII. Beyond this book 1. Toric varieties 2. Newton-Okounkov bodies 3. Bézout problem 4. Newton diagrams 5. Counting real zeroes Appendix A. Commutative algebra results used in chapter III without a proof Appendix B. Miscellaneous commutative algebra 1. Integral domain, UFD, PID 2. Prime and maximal ideals 3. Noetherian rings, Hilbert's basis theorem, annihilators 4. (Algebraic) Field extensions 5. Hilbert's Nullstellensatz 6. Nakayama's lemma 7. Localization, local rings 8. Discrete valuation rings 9. Krull dimension 10. Primary decomposition 11. Length of modules 12. (In)Separable field extensions 13. Rings of formal power series over a field 14. Monomial orders on rings of formal power series 14.1. Monomial orders on rings of formal power series over a field 15. Primitive elements of mathbbZn 16. Symmetric multiadditive functions on a commutative semigroup Appendix C. Some results related to schemes 1. Macaulay's Unmixedness Theorem 2. Properties of order at a point on a possibly non-reduced curve Appendix D. Notation Bibliography