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The invariant theory of binary forms. (English) Zbl 0577.15020
This is a modern exposition of the theory of invariants of binary forms. After introduction, in the second chapter the authors explain the symbolic method of the classical invariant theory. It is based on certain umbral operators on the set of polynomials. Basing on this method, they prove in chapter III the first two fundamental theorems of the invariant theory. Roughly speaking, these theorems assert that any homogeneous covariant of binary forms can be obtained from the values of the umbral operators at the bracket polynomials.
In chapter IV they give an explicit algorithm for expressing in terms of the roots of the binary forms a covariant expressed in umbral notation. This allows to give a short proof of Hermite’s reciprocity law which says that the dimension \(c(n,d,t)\) of the space of covariants of degree \(d\) and order \(t\) of binary forms of degree \(k\) is equal to \(c(d,n,t)\). In chapter 5 the authors explain the theory of apolarity for binary forms. A nice corollary of this theory is the Sylvester theorem on expressing a binary form of odd degree \(n=2k+1\) as a sum of \(k\) \(n\)-th powers of linear forms.
In chapter VI the authors give two proofs of the finiteness theorem. Neither uses Hilbert’s basis theorem. The first is based on the circular straightening algorithm. The second proof is due to Hilbert and uses a combinatorial lemma of Gordon. The article ends with a discussion of some open problems in the invariant theory.

MSC:
15A72 Vector and tensor algebra, theory of invariants
14L24 Geometric invariant theory
14L30 Group actions on varieties or schemes (quotients)
20G05 Representation theory for linear algebraic groups
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