Power Sums, Gorenstein Algebras, and Determinantal Loci

1. Front Matter

   Pages I-XXXI

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2. Forms and catalecticant matrices

   • Anthony Iarrobino, Vassil Kanev

   Pages 3-56

3. Sums of powers of linear forms, and gorenstein algebras

   • Anthony Iarrobino, Vassil Kanev

   Pages 57-72

4. Tangent spaces to catalecticant schemes

   • Anthony Iarrobino, Vassil Kanev

   Pages 73-90

5. The locus PS(s, j; r) of sums of powers, and determinantal loci of catalecticant matrices

   • Anthony Iarrobino, Vassil Kanev

   Pages 91-127

6. Forms and zero-dimensional schemes I: Basic results, and the case r=3

   • Anthony Iarrobino, Vassil Kanev

   Pages 131-205

7. Forms and zero-dimensional schemes, II: Annihilating schemes and reducible Gor(T)

   • Anthony Iarrobino, Vassil Kanev

   Pages 207-236

8. Connectedness and components of the determinantal locus ℙV _s(u, v; r)

   • Anthony Iarrobino, Vassil Kanev

   Pages 237-247

9. Closures of the variety Gor(T), and the parameter space G(T) of graded algebras

   • Anthony Iarrobino, Vassil Kanev

   Pages 249-253

10. Questions and problems

    • Anthony Iarrobino, Vassil Kanev

    Pages 255-264

11. Back Matter

    Pages 265-346

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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.

• Dimension
• Grad
• algebra
• algebraic geometry
• commutative algebra

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