Power Sums, Gorenstein Algebras, and Determinantal Loci
Power Sums, Gorenstein Algebras, and Determinantal Loci

Author: Anthony Iarrobino

Publisher: Springer

Published: 2006-11-14

Total Pages: 365

ISBN-13: 3540467076

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This book treats the theory of representations of homogeneous polynomials as sums of powers of linear forms. The first two chapters are introductory, and focus on binary forms and Waring's problem. Then the author's recent work is presented mainly on the representation of forms in three or more variables as sums of powers of relatively few linear forms. The methods used are drawn from seemingly unrelated areas of commutative algebra and algebraic geometry, including the theories of determinantal varieties, of classifying spaces of Gorenstein-Artin algebras, and of Hilbert schemes of zero-dimensional subschemes. Of the many concrete examples given, some are calculated with the aid of the computer algebra program "Macaulay", illustrating the abstract material. The final chapter considers open problems. This book will be of interest to graduate students, beginning researchers, and seasoned specialists. Prerequisite is a basic knowledge of commutative algebra and algebraic geometry.

Power Sums, Gorenstein Algebras, and Determinantal Loci
Power Sums, Gorenstein Algebras, and Determinantal Loci

Author: Anthony Iarrobino

Publisher: Springer

Published: 2014-03-12

Total Pages: 354

ISBN-13: 9783662214862

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This book treats the theory of representations of homogeneous polynomials as sums of powers of linear forms. The first two chapters are introductory, and focus on binary forms and Waring's problem. Then the author's recent work is presented mainly on the representation of forms in three or more variables as sums of powers of relatively few linear forms. The methods used are drawn from seemingly unrelated areas of commutative algebra and algebraic geometry, including the theories of determinantal varieties, of classifying spaces of Gorenstein-Artin algebras, and of Hilbert schemes of zero-dimensional subschemes. Of the many concrete examples given, some are calculated with the aid of the computer algebra program "Macaulay", illustrating the abstract material. The final chapter considers open problems. This book will be of interest to graduate students, beginning researchers, and seasoned specialists. Prerequisite is a basic knowledge of commutative algebra and algebraic geometry.

Computational Algebraic Geometry
Computational Algebraic Geometry

Author: Hal Schenck

Publisher: Cambridge University Press

Published: 2003-10-06

Total Pages: 212

ISBN-13: 9780521536509

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The interplay between algebra and geometry is a beautiful (and fun!) area of mathematical investigation. Advances in computing and algorithms make it possible to tackle many classical problems in a down-to-earth and concrete fashion. This opens wonderful new vistas and allows us to pose, study and solve problems that were previously out of reach. Suitable for graduate students, the objective of this 2003 book is to bring advanced algebra to life with lots of examples. The first chapters provide an introduction to commutative algebra and connections to geometry. The rest of the book focuses on three active areas of contemporary algebra: Homological Algebra (the snake lemma, long exact sequence inhomology, functors and derived functors (Tor and Ext), and double complexes); Algebraic Combinatorics and Algebraic Topology (simplicial complexes and simplicial homology, Stanley-Reisner rings, upper bound theorem and polytopes); and Algebraic Geometry (points and curves in projective space, Riemann-Roch, Cech cohomology, regularity).

The Art of Doing Algebraic Geometry
The Art of Doing Algebraic Geometry

Author: Thomas Dedieu

Publisher: Springer Nature

Published: 2023-04-14

Total Pages: 421

ISBN-13: 303111938X

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This volume is dedicated to Ciro Ciliberto on the occasion of his 70th birthday and contains refereed papers, offering an overview of important parts of current research in algebraic geometry and related research in the history of mathematics. It presents original research as well as surveys, both providing a valuable overview of the current state of the art of the covered topics and reflecting the versatility of the scientific interests of Ciro Ciliberto.

Recent Advances in Algebraic Geometry
Recent Advances in Algebraic Geometry

Author: Christopher D. Hacon

Publisher: Cambridge University Press

Published: 2015-01-15

Total Pages: 451

ISBN-13: 110764755X

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A comprehensive collection of expository articles on cutting-edge topics at the forefront of research in algebraic geometry.

Deformation of Artinian Algebras and Jordan Type
Deformation of Artinian Algebras and Jordan Type

Author: Anthony Iarrobino

Publisher: American Mathematical Society

Published: 2024-09-06

Total Pages: 254

ISBN-13: 1470473569

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This volume contains the proceedings of the AMS-EMS-SMF Special Session on Deformations of Artinian Algebras and Jordan Type, held July 18?22, 2022, at the Universit‚ Grenoble Alpes, Grenoble, France. Articles included are a survey and open problems on deformations and relation to the Hilbert scheme; a survey of commuting nilpotent matrices and their Jordan type; and a survey of Specht ideals and their perfection in the two-rowed case. Other articles treat topics such as the Jordan type of local Artinian algebras, Waring decompositions of ternary forms, questions about Hessians, a geometric approach to Lefschetz properties, deformations of codimension two local Artin rings using Hilbert-Burch matrices, and parametrization of local Artinian algebras in codimension three. Each of the articles brings new results on the boundary of commutative algebra and algebraic geometry.

Syzygies and Hilbert Functions
Syzygies and Hilbert Functions

Author: Irena Peeva

Publisher: CRC Press

Published: 2007-03-20

Total Pages: 305

ISBN-13: 1420050915

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Hilbert functions and resolutions are both central objects in commutative algebra and fruitful tools in the fields of algebraic geometry, combinatorics, commutative algebra, and computational algebra. Spurred by recent research in this area, Syzygies and Hilbert Functions explores fresh developments in the field as well as fundamental concepts.

Classical Algebraic Geometry
Classical Algebraic Geometry

Author: Igor V. Dolgachev

Publisher: Cambridge University Press

Published: 2012-08-16

Total Pages: 653

ISBN-13: 1139560786

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Algebraic geometry has benefited enormously from the powerful general machinery developed in the latter half of the twentieth century. The cost has been that much of the research of previous generations is in a language unintelligible to modern workers, in particular, the rich legacy of classical algebraic geometry, such as plane algebraic curves of low degree, special algebraic surfaces, theta functions, Cremona transformations, the theory of apolarity and the geometry of lines in projective spaces. The author's contemporary approach makes this legacy accessible to modern algebraic geometers and to others who are interested in applying classical results. The vast bibliography of over 600 references is complemented by an array of exercises that extend or exemplify results given in the book.

Yetter-Drinfel'd Hopf Algebras over Groups of Prime Order
Yetter-Drinfel'd Hopf Algebras over Groups of Prime Order

Author: Yorck Sommerhäuser

Publisher: Springer

Published: 2004-10-19

Total Pages: 161

ISBN-13: 3540454233

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Being the first monograph devoted to this subject, the book addresses the classification problem for semisimple Hopf algebras, a field that has attracted considerable attention in the last years. The special approach to this problem taken here is via semidirect product decompositions into Yetter-Drinfel'd Hopf algebras and group rings of cyclic groups of prime order. One of the main features of the book is a complete treatment of the structure theory for such Yetter-Drinfel'd Hopf algebras.

Algebraic Transformation Groups and Algebraic Varieties
Algebraic Transformation Groups and Algebraic Varieties

Author: Vladimir Leonidovich Popov

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 244

ISBN-13: 3662056526

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The book covers topics in the theory of algebraic transformation groups and algebraic varieties which are very much at the frontier of mathematical research.