BookDOI
Computational methods in commutative algebra and algebraic geometry
TLDR
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.Abstract:
This ACM volume deals with tackling problems that can be represented by data structures which are essentially matrices with polynomial entries, mediated by the disciplines of commutative algebra and algebraic geometry. The discoveries stem from an interdisciplinary branch of research which has been growing steadily over the past decade. 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. Although intended for advanced students and researchers with interests both in algebra and computation, many parts may be read by anyone with a basic abstract algebra course.read more
Citations
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Book
A Singular Introduction to Commutative Algebra
TL;DR: A singular introduction to commutative algebra as mentioned in this paper is one of the most widely used works in algebraic geometry, with a broad coverage of theoretical topics in the portions of the algebra closest to algebraic geometrical geometry.
Posted Content
On the integral closure of ideals
TL;DR: In this paper, the authors show that in a broad class of ideals their radicals play a fundamental role in testing for integral closedness, and in case $I
eq \bar{I}$, $\surd{I]$ is still helpful in finding some fresh new elements in $\bar{ I}\setminus I$.
Journal ArticleDOI
Alexander Duality for Stanley–Reisner Rings and Squarefree Nn-Graded Modules
TL;DR: In this article, it was shown that the i-linear strand of the minimal free resolution of a Stanley-Reisner ideal IΔ⊂ S has the same information as the module structure of ExtiS(k[Δ∨,], εS), where Δ∨ is the Alexander dual of Δ.
BookDOI
Edge Ideals: Algebraic and Combinatorial Properties
Susan Morey,Rafael H. Villarreal +1 more
TL;DR: A survey of algebraic and combinatorial properties of R/I(C) and C, respectively, can be found in this paper, where the authors give a criterion to estimate the regularity of the ideal of vertex covers of C. They also examine the associated primes of powers of edge ideals and show that these sets form an ascending chain.
Journal ArticleDOI
Factorization-free decomposition algorithms in differential algebra
TL;DR: Insight on the structure of differential ideals defined by coherent autoreduced set allows one to uncouple the differential and algebraic computations in a decomposition algorithm and an effective version of Ritt's algorithm can be simply described.
References
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Book
Commutative Algebra: with a View Toward Algebraic Geometry
TL;DR: In this article, the authors define basic constructions and dimension theory, and apply them to the problem of homological methods for combinatorial problem solving in the context of homology.
Book
Introduction to Commutative Algebra
TL;DR: It is shown here how the Noetherian Rings and Dedekind Domains can be transformed into rings and Modules of Fractions using the following structures:
Book
The Mathematica Book
TL;DR: Mathematica has defined the state of the art in technical computing for over a decade, and has become a standard in many of the world's leading companies and universities as discussed by the authors.
BookDOI
Ideals, Varieties, and Algorithms
TL;DR: In the Groebner package, the most commonly used commands are NormalForm, for doing the division algorithm, and Basis, for computing a Groebners basis as mentioned in this paper. But these commands require a large number of variables.
Book
Introduction to Homological Algebra
TL;DR: The landscape of homological algebra has evolved over the last half-century into a fundamental tool for the working mathematician as discussed by the authors, which is suitable for second or third year graduate students.