Classical Invariant TheoryThere 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. |
Contents
PreludeQuadratic Polynomials | 1 |
Basic Invariant Theory for Binary Forms | 11 |
Homogeneous Functions and Forms | 19 |
The Simplest Examples | 27 |
Joint Covariants and Polarization | 33 |
The Hilbert Basis Theorem | 39 |
Transvectants | 86 |
Symbolic Methods | 99 |
Graphical Methods | 128 |
Lie Groups and Moving Frames | 150 |
Infinitesimal Methods | 198 |
Multivariate Polynomials | 228 |
| 247 | |
| 260 | |
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Common terms and phrases
analytic atom basic binary form bracket factors bracket monomial bracket polynomial canonical forms classical invariant theory coefficients complex computation contravariant coordinates corresponding covariant covariant of weight defined determinant differential equations differential invariants differential operators differential polynomial digraph discriminant equivalent Euclidean group example Exercise form of degree form Q formula Fundamental Theorem G-invariant geometry GL(n group action hence Hessian Hilbert basis homogeneous functions homogeneous polynomial identity induced infinitesimal inhomogeneous invariant theory inverse irreducible Jacobian joint invariant Lie algebra Lie group linear combination linear group linear transformations matrix method monomial multiple nonzero omega process one-parameter subgroup orbits parameters partial transvectant polynomial covariant projective proof Prove quadratic forms quadratic polynomial quartic relative invariant Remark represents result root signature curve subset subspace symbolic form symbolic letters symmetric symmetry group syzygy tensor tion transformation group transformation rules transvectant unimodular valence vanishes