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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.
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