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Power Sums, Gorenstein Algebras, and Determinantal Loci : Lecture Notes in Artificial Intelligence - Anthony A. Iarrobino

Power Sums, Gorenstein Algebras, and Determinantal Loci

Lecture Notes in Artificial Intelligence

Paperback Published: January 2000
ISBN: 9783540667667
Number Of Pages: 354

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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 & focus on binary forms & 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 & algebraic geometry, including the theories of determining varieties, of classifying spaces of Gorenstein-Artin algebra & 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 & seasoned specialists. Prerequisite is a basic knowledge of commutative algebra & algebraic geometry.

Introduction: Informal History and Brief Outlinep. xiii
Canonical forms, and catalecticant matrices of higher partial derivatives of a formp. xiii
Apolarity and Artinian Gorenstein algebrasp. xviii
Families of sets of pointsp. xxi
Brief summary of chaptersp. xxii
Catalecticant Varietiesp. 1
Forms and Catalecticant Matricesp. 3
Apolarity and catalecticant varieties: the dimensions of the vector spaces of higher partialsp. 3
Determinantal loci of the first catalecticant, the Jacobianp. 16
Binary forms and Hankel matricesp. 22
Detailed summary and preparatory resultsp. 41
Sums of Powers of Linear Forms, and Gorenstein Algebrasp. 57
Waring's problem for general formsp. 57
Uniqueness of additive decompositionsp. 62
The Gorenstein algebra of a homogeneous polynomialp. 67
Tangent Spaces to Catalecticant Schemesp. 73
The tangent space to the determinantal scheme Vs(u,v;r) of the catalecticant matrixp. 73
The tangent space to the scheme Gor(T) parametrizing forms with fixed dimensions of the partialsp. 79
The Locus PS(s,j;r) of Sums of Powers, and Determinantal Loci of Catalecticant Matricesp. 91
The case r = 3p. 92
Sets of s points in Pr-1 and Gorenstein idealsp. 102
Gorenstein ideals whose lowest degree generators are a complete intersectionp. 108
The smoothness and dimension of the scheme Gor(T) when r = 3: a surveyp. 116
Catalecticant Varieties and the Punctual Hilbert Schemep. 129
Forms and Zero-Dimensional Schemes I: Basic Results, and the Case r = 3p. 131
Annihilating scheme in Pr-1 of a formp. 135
Flat families of zero-dimensional schemes and limit idealsp. 142
Existence theorems for annihilating schemes when r = 3p. 150
The generator and relation strata of the variety Gor(T) parametrizing Gorenstein algebrasp. 151
The morphism from Gor(T): the case T ⊃ D(s,s,s)p. 156
Morphism: the case T ⊃ D(s - a,s,s,s - a)p. 167
Morphism: the case T ⊃ D(s - a,s,s - a)p. 172
Adimension formula for the variety Gor(T)p. 179
Power sum representations in three and more variablesp. 182
Betti strata of the punctual Hilbert schemep. 189
The length of a form, and the closure of the locus PS(s,j;3) of power sumsp. 197
Codimension three Gorenstein schemes in Pnp. 201
Forms and Zero-Dimensional Schemes, II: Annihilating Schemes and Reducible Gor(T)p. 207
Uniqueness of the annihilating scheme; closure of PS(s,J;r)p. 208
Varieties Gor(T), T = T(j,r), with several componentsp. 214
Other reducible varieties Gor(T)p. 224
Locally Gorenstein annihilating schemesp. 226
Connectedness and Components of the Determinantal Locus PVs(u,v;r)p. 237
Connectedness of PVs(u,v;r)p. 237
The irreducible components of Vs(u,v;r)p. 241
Multisecant varieties of the Veronese varietyp. 245
Closures of the Variety Gor(T), and the Parameter Space G(T) of Graded Algebrasp. 249
Questions and Problemsp. 255
Divided Power Rings and Polynomial Ringsp. 265
Height Three Gorenstein Idealsp. 271
Pfaffian formulasp. 272
Resolutions of height 3 Gorenstein ideals andtheir squaresp. 276
Resolutions of annihilating ideals of power sumsp. 280
Maximum Betti numbers, given Tp. 282
The Gotzmann Theorems and the Hilbert Scheme (Anthony Iarrobino and Steven L. Kleiman)p. 289
Order sequences and Macaulay's Theorem on Hilbert functionsp. 290
Macaulay and Gotzmann polynomialsp. 293
Gotzmann's Persistence Theorem and m-Regularityp. 297
The Hilbert scheme HilbP (Pr-1)p. 302
Gorenstein sequences having a subsequence of maximal growth, and HilbP (Pr-1)p. 307
Examples of "Macaulay" Scriptsp. 313
Concordance with the 1996 Versionp. 317
Referencesp. 319
Indexp. 335
Index of Namesp. 341
Index of Notationp. 343
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540667667
ISBN-10: 3540667660
Series: Lecture Notes in Artificial Intelligence
Audience: General
Format: Paperback
Language: English
Number Of Pages: 354
Published: January 2000
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6  x 2.03
Weight (kg): 0.54

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