# Computational Algebraic Geometry

@inproceedings{Schenck2003ComputationalAG, title={Computational Algebraic Geometry}, author={Hal Schenck}, year={2003} }

Preface 1. Basics of commutative algebra 2. Projective space and graded objects 3. Free resolutions and regular sequences 4. Groebner bases 5. Combinatorics and topology 6. Functors: localization, hom, and tensor 7. Geometry of points 8. Homological algebra, derived functors 9. Curves, sheaves and cohomology 10. Projective dimension A. Abstract algebra primer B. Complex analysis primer Bibliography.

## 97 Citations

Algebraic Geometry: An Introduction

- Mathematics
- 2007

Affine algebraic sets.- Projective algebraic sets.- Sheaves and varieties.- Dimension.- Tangent spaces and singular points.- Bezout's theorem.- Sheaf cohomology.- Arithmetic genus of curves and the…

Monomial Algebras

- Mathematics
- 2015

Bringing together several areas of pure and applied mathematics, this book shows how monomial algebras are related to polyhedral geometry, combinatorial optimization, and combinatorics of…

Combinatorial Koszul Homology: Computations and Applications

- Mathematics
- 2008

With a particular focus on explicit computations and applications of the Koszul homology and Betti numbers of monomial ideals, the main goals of this thesis are the following:
Analyze the Koszul…

homalg – A META-PACKAGE FOR HOMOLOGICAL ALGEBRA

- Mathematics, Computer Science
- 2008

The central notion of this work is that of a functor between categories of finitely presented modules over so-called computable rings, i.e. rings R where one can algorithmically solve inhomogeneous…

A Survey of Stanley–Reisner Theory

- Mathematics
- 2014

We survey the Stanley–Reisner correspondence in combinatorial commutative algebra, describing fundamental applications involving Alexander duality, associated primes, f- and h-vectors, and Betti…

A Case Study in Bigraded Commutative Algebra

- Mathematics
- 2007

We study the commutative algebra of three bihomogeneous polynomials p_0,p_1,p_2 of degree (2,1) in variables x,y;z,w, assuming that they never vanish simultaneously on P^1 x P^1. Unlike the situation…

On minimal free resolutions of sub-permanents and other ideals arising in complexity theory

- MathematicsJournal of Algebra
- 2018

Homological algebra and problems in combinatorics and geometry

- Mathematics
- 2007

Homological Algebra and problems in Combinatorics and Geometry. (May 2007) Ştefan Ovidiu Tohǎneanu, B.S., University of Bucharest; M.S. (Algebra), University of Bucharest; M.S. (Analysis), University…

Heterotic Compactification, An Algorithmic Approach

- Mathematics
- 2007

We approach string phenomenology from the perspective of computational algebraic geometry, by providing new and efficient techniques for proving stability and calculating particle spectra in…

Polynomial Interpolation in Higher Dimension: From Simplicial Complexes to GC Sets

- MathematicsSIAM J. Numer. Anal.
- 2017

This work investigates GC sets in dimension three or more, and shows that one way to obtain such sets is from the combinatorics of simplicial complexes.

## References

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Preface Introduction Conventions and terminology 1. Commutative rings and modules 2. prime ideals 3. Properties of extension rings 4. Valuation rings 5. Dimension theory 6. Regular sequences 7.…

Elements of algebraic topology

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Elements of Algebraic Topology provides the most concrete approach to the subject. With coverage of homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in…

Commutative Algebra: with a View Toward Algebraic Geometry

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Introduction.- Elementary Definitions.- I Basic Constructions.- II Dimension Theory.- III Homological Methods.- Appendices.- Hints and Solutions for Selected Exercises.- References.- Index of…

An introduction to Gröbner bases

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This book discusses rings, Fields, and Ideals, and applications of Grobner Bases, as well as improvements to Buchberger's Algorithm and other topics.

Methods for computing in algebraic geometry and commutative algebra Rome, March 1990

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This chapter discusses the development of Grobner bases, computer algebra systems which are readily available to researchers, and the great increase in power of computers in this last decade.

Cohomology of Vector Bundles and Syzygies

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- 2003

1. Introduction 2. Schur functions and Schur complexes 3. Grassmannians and flag varieties 4. Bott's theorem 5. The geometric technique 6. The determinantal varieties 7. Higher rank varieties 8. The…

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A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications…

Gröbner bases and convex polytopes

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Grobner basics The state polytope Variation of term orders Toric ideals Enumeration, sampling and integer programming Primitive partition identities Universal Grobner bases Regular triangulations The…

Computational methods in commutative algebra and algebraic geometry

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The author covers a wide range, from showing how to obtain deep heuristics in a computation of a ring, a module or a morphism, to developing means of solving nonlinear systems of equations - highlighting the use of advanced techniques to bring down the cost of computation.