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 unimodular transformations of the variables x and y.

1 Terminology
2 Examples
3 The ring of invariants
4 Invariants of a ternary cubic
5 References

Terminology

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 are 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, octavics, nonics, and decimics. "Quantic" is an old name for a form of arbitrary degree. Forms in 1, 2, 3, ... variables are called unary, binary, ternary, ... forms.

Examples

A form f is itself is 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

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

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

The catalecticant is an invariant of a form of even degree.

The Jacobian

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

is a simultaneous invariant of two forms f, g.

The ring of invariants

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.

For linear forms ax + by the only invariants are constants. The algebra of covariants is generated by the form itself of degree 1 and order 1.
The algebra of invariants of the quadratic form ax² + 2bxy + cy² is a polynomial algebra in 1 variable generated by the discriminant b² − ac of degree 2. The algebra of covariants is a polynomial algebra in 2 variables generated by the discriminant together with the form f itself (of degree 1 and order 2). (Schur 1968, II.8) (Hilbert 1993, XVI, XX)
The algebra of invariants of the cubic form ax³ + 3bx²y + 3cxy² + dy³ is a polynomial algebra in 1 variable generated by the discriminant D = 3b²c² + 6abcd − 4b³d − 4c³a − a²d² of degree 4. The algebra of covariants is generated by the discriminant, the form itself (degree 1, order 3), the Hessian H (degree 2, order 2) and a covariant T of degree 3 and order 3. They are related by the syzygy 4h³=Df²-T² of degree 6 and order 6. (Schur 1968, II.8) (Hilbert 1993, XVII, XX)
The algebra of invariants of a quartic form is generated by invariants i, j of degrees 2, 3. The algebra of covariants is generated by these two invariants together with the form f of degree 1 and order 4, the Hessian H of degree 2 and order 4, and a covariant T of degree 3 and order 6. They are related by a syzygy jf³−Hf²i+4H³+T²=0 of degree 6 and order 12. (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 generators of degrees 4, 8, 12 generate a polynomial ring, which contains the square of the generator of degree 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. (Schur 1968, II.9) (Hilbert 1993, XVIII)
The algebra of invariants of a sextic form is generated by invariants of degree 2, 4, 6, 10, 15. The generators of degrees 2, 4, 6, 10 generate a polynomial ring, which contains the square of the generator of degree 15. (Schur 1968, II.9)
von Gall (1888) and Dixmier & Lazard (1986) showed that the algebra of invariants of a degree 7 form is generated by a set with 1 invariant of degree 4, 3 of degree 8, 6 of degree 12, 4 of degree 14, 2 of degree 16, 9 of degree 18, and one of each of the degrees 20, 22, 26, 30
von Gall (1880) and Shioda (1967) showed that the algebra 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 ideal of relations between them is generated by elements of degrees 16, 17, 18, 19, 20.
Brouwer & Popoviciu (2010a) showed that the algebra of invariants of a degree 9 form is generated by 92 invariants
Brouwer & Popoviciu (2010b) showed that the algebra of invariants of a degree 10 form is generated by 106 invariants

The number of generators for invariants and covariants of binary forms can be found in (sequence A036983 in OEIS) and (sequence A036984 in OEIS), respectively.

Invariants of a ternary cubic

The algebra of invariants of a ternary cubic under SL₃(C) is a polynomial algebra generated by two invariants of degrees 4 and 6. The invariants are rather complicated, and are given explicitly in (Sturmfels 1993, 4.4.7, 4.5.3)

References

Brouwer, Andries E.; Popoviciu, Mihaela (2010a), "The invariants of the binary nonic", Journal of Symbolic Computation 45 (6): 709–720, doi:10.1016/j.jsc.2010.03.003, ISSN 0747-7171, MR 2639312
Brouwer, Andries E.; Popoviciu, Mihaela (2010b), "The invariants of the binary decimic", Journal of Symbolic Computation 45 (8): 837–843, doi:10.1016/j.jsc.2010.03.002, ISSN 0747-7171, MR 2657667
Dixmier, Jacques; Lazard, D. (1988), "Minimum number of fundamental invariants for the binary form of degree 7", Journal of Symbolic Computation 6 (1): 113–115, doi:10.1016/S0747-7171(88)80026-9, ISSN 0747-7171, MR 961375
Gall, F. von (1880), "Das vollständige Formensystem einer binären Form achter Ordnung", Mathematische Annalen 17 (1): 31–51, doi:10.1007/BF01444117, ISSN 0025-5831, MR 1510048
Gall, Frhr v. (1888), "Das vollständige Formensystem der binären Form 7^terOrdnung", Mathematische Annalen 31 (3): 318–336, doi:10.1007/BF01206218, ISSN 0025-5831, MR 1510486
Gordan, Paul (1868), "Beweis, dass jede Covariante und Invariante einer binären Form eine ganze Funktion mit numerischen Coeffizienten einer endlichen Anzahl solcher Formen ist", J. F. Math 69 (69): 323–354, doi:10.1515/crll.1868.69.323
Hilbert, David (1993) [1897], Theory of algebraic invariants, Cambridge University Press, ISBN 978-0-521-44457-6, MR 1266168, http://books.google.com/books?isbn=0521449030
Kung, Joseph P. S.; Rota, Gian-Carlo (1984), "The invariant theory of binary forms", American Mathematical Society. Bulletin. New Series 10 (1): 27–85, doi:10.1090/S0273-0979-1984-15188-7, ISSN 0002-9904, MR 722856, http://www.ams.org/journals/bull/1984-10-01/S0273-0979-1984-15188-7
Schur, Issai (1968), Grunsky, Helmut, ed., Vorlesungen über Invariantentheorie, Die Grundlehren der mathematischen Wissenschaften, 143, Berlin, New York: Springer-Verlag, ISBN 978-3540041399, MR 0229674
Shioda, Tetsuji (1967), "On the graded ring of invariants of binary octavics", American Journal of Mathematics 89 (4): 1022–1046, doi:10.2307/2373415, ISSN 0002-9327, JSTOR 2373415, MR 0220738
Sturmfels, Bernd (1993), Algorithms in invariant theory, Texts and Monographs in Symbolic Computation, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-211-77417-5, ISBN 978-3-211-82445-0, MR 1255980
Sylvester, J. J.; Franklin, F. (1879), "Tables of the Generating Functions and Groundforms for the Binary Quantics of the First Ten Orders", American Journal of Mathematics 2 (3): 223–251, doi:10.2307/2369240, ISSN 0002-9327, MR 1505222