Power Sums, Gorenstein Algebras, and Determinantal Loci

Anthony Iarrobino, Vassil Kanev (auth.)3540667660, 9783540667667

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.

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Table of contents :
Forms and catalecticant matrices….Pages 3-56
Sums of powers of linear forms, and gorenstein algebras….Pages 57-72
Tangent spaces to catalecticant schemes….Pages 73-90
The locus PS(s, j; r) of sums of powers, and determinantal loci of catalecticant matrices….Pages 91-127
Forms and zero-dimensional schemes I: Basic results, and the case r =3….Pages 131-205
Forms and zero-dimensional schemes, II: Annihilating schemes and reducible Gor(T) ….Pages 207-236
Connectedness and components of the determinantal locus ℙ V s ( u, v; r )….Pages 237-247
Closures of the variety Gor(T) , and the parameter space G(T) of graded algebras….Pages 249-253
Questions and problems….Pages 255-264