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Power sums, Gorenstein algebras, and determinantal loci. With an appendix ‘The Gotzmann theorems and the Hilbert scheme’ by Anthony Iarrobino and Steven L. Kleiman. (English) Zbl 0942.14026
Lecture Notes in Mathematics. 1721. Berlin: Springer. xxxi, 345 p. (1999).
This book is concerned with the classical problem of representing a homogeneous polynomial (shortly a form) as a sum of powers of linear forms. This problem is related to the study of the loci of the forms of given degree with a given sequence of dimensions for the spaces generated by their order-\(i\) higher partial derivatives. These dimensions are the ranks of the catalecticant matrices whose columns are the coefficients of the partial derivatives of the form hence the above mentioned loci are of a determinantal nature. Moreover, the set of polynomials apolar to the given form (called a principal system by F. H. S. Macaulay) is a homogeneous ideal of the polynomial ring and the quotient algebra is a graded Gorenstein Artin algebra whose Hilbert function is exactly the above sequence of ranks. Finally, the main problem and the study of the catalecticant determinantal loci are also related to the study of the punctual Hilbert schemes of projective spaces. This is one of the more technically involved topics of the book and is investigated in part II.
In the case of quadratic forms (degree = 2) the main problem is solved by standard linear algebra and the case of binary forms (2 indeterminates) was solved by J. Sylvester in the middle of the 19th century. This classical material is explained in the first two chapters of the book. The second chapter also reports on the solution of Waring’s problem (in any number of indeterminates) which asks for the minimum integer \(s\) such that a general form of degree \(j\) can be represented as the sum of the \(j\)-th powers of \(s\) linear forms. This solution has been obtained by J. Alexander and A. Hirschowitz [J. Algebr. Geom. 4, 201-222 (1995; Zbl 0829.14002)].
Chapters 3-8 contain the new results of the authors which, together with the results of S. J. Diesel [Pac. J. Math. 172, No. 2, 365-397 (1996; Zbl 0882.13021)], J. O. Kleppe [J. Algebra 200, No. 2, 606-628 (1998; Zbl 0928.14005)], M. Boij [Bull. Lond. Math. Soc. 31, No. 1, 11-16 (1999)], A. Conca and G. Valla [Math. Z. 230, No. 4, 753-784 (1999; Zbl 0927.13020)] and others, largely clarify the case of 3 indeterminates. A powerful tool in this case is the structure theorem of height 3 Gorenstein ideals of D. A. Buchsbaum and D. Eisenbud [Am. J. Math. 99, 447-485 (1977; Zbl 0373.13006)].
The book also contains a large number of comments, (counter-)examples and surveys of recent results illustrating the new pheomena that occur in the case of more than 3 indeterminates. The last chapter (the 9th) is a list of open problems. There are several useful appendices: one about divided powers (which are needed if one wants to work in arbitrary characteristic), one about height 3 Gorenstein idelas and another one, written by A. Iarrobino and S. L. Kleiman, about Gotzmann theorems and the Hilbert schemes.
Since a preliminary version of the book circulated as a preprint from 1996, the authors include an appendix in which they specify the changes occurring in the published version. – Concluding, this book is a detailed and stimulating overview of a fascinating current research topic having classical roots and a long history.

MSC:
14M12 Determinantal varieties
14C05 Parametrization (Chow and Hilbert schemes)
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra
13C40 Linkage, complete intersections and determinantal ideals
14N99 Projective and enumerative algebraic geometry
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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