http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
Power sums, Gorenstein algebras, and determinantal loci
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.
Introduction: Informal History and Brief Outline -- Catalecticant Varieties: Forms and Catalecticant Matrices. Sums of Powers and Linear Forms, and Gorenstein Algebras. Tangent Spaces to Catalecticant Schemes. The Locus PS(s, j;r) of Sums of Powers, and Determinantal Loci of Catalecticant Matrices -- Catalecticant Varieties and the Punctual Hilbert Scheme: Forms and Zero-Dimensional SchemesI: Basic Results, and the Case r = 3. Forms and Zero-Dimensional Schemes, II: Annihilating Schemes and Reducible Gor(T). Connectedness and Components of the Determinantal Locus PVs(u, v;r). Closures of the Variety Gor(T), and the Parameter Space G(T) of Graded Algebras -- Questions and Problems -- Appendix A: Divided Rings and Polynomial Rings -- Appendix B: Height Three Gorenstein Ideals -- Appendix C: The Gotzmann Theorems and the Hilbert Scheme (Anthony Iarrobino and S.L. Kleiman) -- Appendix D: Exemples of "Macaulay" Scripts -- Appendix E: Concordance with the 1996 Version.
"With an appendix the Gotzmann Theorems and the Hilbert Scheme by Anthony Iarrobino and Steven L. Kleiman."
Includes bibliographical references (pages [319]-334) and indexes.
Iarrobino, Anthony A. (Anthony Ayers), 1943-
Lecture notes in mathematics (Springer-Verlag) ; 1721