Invariant of a binary form

In mathematical invariant theory, an invariant of a binary form is a polynomial in the coefficients of a binary form in two variables x and y that remains invariant under the special linear group acting on the variables x and y.

A binary form (of degree n) is a homogeneous polynomial Σn
i=0 (n
i)a_n−ixⁿ⁻ⁱyⁱ = a_nxⁿ + (n
1)a_n−1xⁿ⁻¹y + ... + a₀yⁿ. The group SL₂(C) acts on these forms by taking x to ax + by and y to cx + dy. This induces an action on the space spanned by a₀, ..., a_n and on the polynomials in these variables. An invariant is a polynomial in these n + 1 variables a₀, ..., a_n that is invariant under this action. More generally a covariant is a polynomial in a₀, ..., a_n, x, y that is invariant, so an invariant is a special case of a covariant where the variables x and y do not occur. More generally still, a simultaneous invariant is a polynomial in the coefficients of several different forms in x and y.

In terms of representation theory, given any representation V of the group SL₂(C) one can ask for the ring of invariant polynomials on V. Invariants of a binary form of degree n correspond to taking V to be the (n + 1)-dimensional irreducible representation, and covariants correspond to taking V to be the sum of the irreducible representations of dimensions 2 and n + 1.

The invariants of a binary form form a graded algebra, and Gordan (1868) proved that this algebra is finitely generated if the base field is the complex numbers.

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