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On the complexity of counting irreducible components and computing Betti numbers of algebraic varieties / Peter Scheiblechner. 2007
Content
Introduction
General Upper Bounds
Lower Complexity Bounds
Fixing Parameters
Outline
Credits
Preliminaries
Algebraic Geometry
Differential Forms
Models of Computation
Structural Complexity
Efficient Parallel Algorithms
Squarefree Regular Chains
I Upper Bounds
Transfer Results
Transfer Results for Complexity Classes
Generic and Randomised Reductions
Counting Connected Components
The Zeroth de Rham Cohomology
Modified Pseudo Remainders
Computing Differentials
Proof of Theorem 3.1
Counting Irreducible Components
Affine vs. Projective Case
Locally Constant Rational Functions
Proof of Theorem 4.1
Hilbert Polynomial
Bound for the Index of Regularity
Computing the Hilbert Polynomial
II Lower Bounds
Connectedness
Basic Notations
Obtaining an Acyclic Configuration Graph
Embedding the Configuration Graph
Equations for the Embedded Graph
Proof of Theorem 6.1
Appendix. The Real Reachability Problem
Betti Numbers
The Affine Case
The Projective Case
III Fixing Parameters
Counting Irreducible Factors
Cohomology of a Hypersurface Complement
Structure Theorem for Closed 1-Forms
Proof of Theorem 8.3
Characterising Exact Forms
Proof of Theorem 8.1
Counting Irreducible Components Revisited
Fixed Number of Equations
Proof of the Main Results
Transversality
Explicit Genericity Condition for Bertini
Expressing the Genericity Condition
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