Invariant of a binary form

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.

Forms of degrees 2, 3, 4, 5, 6, 7, 8, 9, 10 are sometimes called quadrics, cubic, quartics, quintics, sextics, septics or septimics, octics or octavics, nonics, and decics or decimics. "Quantic" is an old name for a form of arbitrary degree. Forms in 1, 2, 3, 4, ... variables are called unary, binary, ternary, quaternary, ... forms.

A form f is itself a covariant of degree 1 and order n.

The discriminant of a form is an invariant.

The resultant of two forms is a simultaneous invariant of them.

The Hessian covariant of a form Hilbert (1993, p.88) is the determinant of the Hessian matrix

${\displaystyle H(f)={\begin{bmatrix}{\frac {\partial ^{2}f}{\partial x^{2}}}&{\frac {\partial ^{2}f}{\partial x\,\partial y}}\\[10pt]{\frac {\partial ^{2}f}{\partial y\,\partial x}}&{\frac {\partial ^{2}f}{\partial y^{2}}}\end{bmatrix}}.}$

It is a covariant of order 2n− 4 and degree 2.

The catalecticant is an invariant of degree n/2+1 of a binary form of even degree n.

The canonizant is a covariant of degree and order (n+1)/2 of a binary form of odd degree n.

The Jacobian

${\displaystyle \det {\begin{bmatrix}{\frac {\partial f}{\partial x}}&{\frac {\partial f}{\partial y}}\\[10pt]{\frac {\partial g}{\partial x}}&{\frac {\partial g}{\partial y}}\end{bmatrix}}}$

is a simultaneous invariant of two forms f, g.

The structure of the ring of invariants has been worked out for small degrees. Sylvester & Franklin (1879) gave tables of the numbers of generators of invariants and covariants for forms of degree up to 10, though the tables have a few minor errors for large degrees, mostly where a few invariants or covariants are omitted.

The covariants of a binary form are essentially the same as joint invariants of a binary form and a binary linear form. More generally, on can ask for the joint invariants (and covariants) of any collection of binary forms. Some cases that have been studied are listed below.

• Ternary cubic
• Ternary quartic

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• Brouwer, Andries E., Invariants of binary forms