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

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