Classical Invariant Theory
There has been a resurgence of interest in classical invariant theory driven by several factors: new theoretical developments; a revival of computational methods coupled with powerful new computer algebra packages; and a wealth of new applications, ranging from number theory to geometry, physics to computer vision. This book provides readers with a self-contained introduction to the classical theory as well as modern developments and applications. The text concentrates on the study of binary forms (polynomials) in characteristic zero, and uses analytical as well as algebraic tools to study and classify invariants, symmetry, equivalence and canonical forms. A variety of innovations make this text of interest even to veterans of the subject; these include the use of differential operators and the transform approach to the symbolic method, extension of results to arbitrary functions, graphical methods for computing identities and Hilbert bases, complete systems of rationally and functionally independent covariants, introduction of Lie group and Lie algebra methods, as well as a new geometrical theory of moving frames and applications. Aimed at advanced undergraduate and graduate students the book includes many exercises and historical details, complete proofs of the fundamental theorems, and a lively and provocative exposition.
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Prelude Quadratic Polynomials
Basic Invariant Theory for Binary Forms
Groups and Transformations
Representations and Invariants
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according action acts algebra analytic apply atom basic basis binary form bracket bracket factors canonical forms Chapter classical coefficients combination complex components computation connected Consider consists constant construct coordinates corresponding covariant cubic curve defined Definition depending determinant differential differential invariants digraph direct discriminant element equals equation equivalent Euclidean Example Exercise fact formula function fundamental geometry given GL(n hence Hessian Hilbert homogeneous homogeneous functions identity implies important induced infinitesimal invariant invariant theory joint known leading Lie group linear matrix meaning method monomial multiple normalization Note obtained orbits ordinary original partial particular polynomial projective proof Prove quadratic quartic reducible Remark representation represents respect result root rules signature single solution space subgroup symbolic symmetric syzygy Theorem theory tion transformation group transvectant vanishes variables vector weight
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