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
More generally still, a simultaneous invariant is a polynomial in the coefficients of several different forms in
-dimensional irreducible representation, and covariants correspond to taking
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.
The Hessian covariant of a form Hilbert (1993, p.88) is the determinant of the Hessian matrix It is a covariant of order 2n− 4 and degree 2.
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 algebra of covariants is generated by the form itself of degree 1 and order 1.
is a polynomial algebra in 1 variable generated by the discriminant
(Schur 1968, II.8) (Hilbert 1993, XVI, XX) The algebra of invariants of the cubic form
is a polynomial algebra in 1 variable generated by the discriminant
The algebra of covariants is generated by the discriminant, the form itself (degree 1, order 3), the Hessian
This ring is naturally isomorphic to the ring of modular forms of level 1, with the two generators corresponding to the Eisenstein series
The algebra of covariants is generated by these two invariants together with the form
(Schur 1968, II.8) (Hilbert 1993, XVIII, XXII) The algebra of invariants of a quintic form was found by Sylvester and is generated by invariants of degree 4, 8, 12, 18.
The invariants are rather complicated to write out explicitly: Sylvester showed that the generators of degrees 4, 8, 12, 18 have 12, 59, 228, and 848 terms often with very large coefficients.
The ring of invariants is closely related to the moduli space of curves of genus 2, because such a curve can be represented as a double cover of the projective line branched at 6 points, and the 6 points can be taken as the roots of a binary sextic.
The ring of invariants of binary septics is anomalous and has caused several published errors.
Cayley claimed incorrectly that the ring of invariants is not finitely generated.
Sylvester & Franklin (1879) gave lower bounds of 26 and 124 for the number of generators of the ring of invariants and the ring of covariants and observed that an unproved "fundamental postulate" would imply that equality holds.
However von Gall (1888) showed that Sylvester's numbers are not equal to the numbers of generators, which are 30 for the ring of invariants and at least 130 for the ring of covariants, so Sylvester's fundamental postulate is wrong.
Sylvester & Franklin (1879) showed that the ring of invariants of a degree 8 form is generated by 9 invariants of degrees 2, 3, 4, 5, 6, 7, 8, 9, 10, and the ring of covariants is generated by 69 covariants.
August von Gall (von Gall (1880)) and Shioda (1967) confirmed the generators for the ring of invariants and showed that the ideal of relations between them is generated by elements of degrees 16, 17, 18, 19, 20.
Cröni, Hagedorn, and Brouwer[1] computed 476 covariants, and Lercier & Olive showed that this list is complete.
Sylvester stated that the ring of invariants of binary decics is generated by 104 invariants the ring of covariants by 475 covariants; his list is to be correct for degrees up to 16 but wrong for higher degrees.
Hagedorn and Brouwer[1] computed 510 covariants, and Lercier & Olive showed that this list is complete.
The ring of invariants of binary forms of degree 11 is complicated and has not yet been described explicitly.
The number of generators for invariants and covariants of binary forms can be found in (sequence A036983 in the OEIS) and (sequence A036984 in the OEIS), respectively.
More generally, on can ask for the joint invariants (and covariants) of any collection of binary forms.