PREFACE to "Power Sums, Gorenstein Algebra, and Determinantal Varieties"
This book is devoted to a classical problem with a long history --- that of representing a homogeneous polynomial as a sum of powers of linear forms. This problem is closely related to another interesting topic --- the study of the loci which parametrize homogeneous polynomials with a given sequence of dimensions for the spaces spanned by their order-$i$ higher partial derivatives. Here a convenient tool to work with are the catalecticant matrices associated to a homogeneous polynomial, whose columns are the coefficients of its partial derivatives in appropriate monomial bases --- the above dimensions are then the ranks of the catalecticant matrices, and the above parametric varieties are their determinantal loci.

In the Introduction we define all basic notions in an informal way, in the classical setting of characteristic zero. We hope this will facilitate reading the book, where setting of arbitrary characteristic is adopted. Our experience has been that with a little more effort almost all results valid in characteristic zero can be extended to arbitrary characteristic, replacing the ring of polynomials by the ring of divided powers; the two rings are isomorphic when the characteristic is zero.

The first two chapters are mainly expository and are intended to give an account of what was already known about catalecticant matrices, especially those associated with the first partial derivatives or with homogeneous polynomials in two variables. We aimed to make this part of the book as self-contained as possible, and include full proofs of some material scattered in the literature or contained in some hardly available old books. Chapters 3 -- 8 as well as Section 2.2 of Chapter 2 contain our new results on the subject. We also included Sections 4.4 and 6.4 in which some recent development due to various authors is surveyed --- a development partially inspired by earlier preliminary versions of this memoir circulated in 1995 -- 1996 \cite{IK}.

The expert already familiar with the basic notions and notation may wish to skip to the Brief Summary of Chapters at the end of the Introduction, then skim the expository part of the first two chapters, noting especially the Detailed Summary (Section~\ref{1C}), before looking for topics of particular interest




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Last modified: January 27, 2000. URL:http://www.math.neu.edu/~iarrobino/pref.html.