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.

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. The invariants are polynomials in these n+1 variables a₀, ..., a_n that are invariant under this action. 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.

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.

1. For linear forms ax + by the only invariants are constants.
2. The algebra of invariants of the quadratic form ax² + 2bxy + cy² is a polynomial algebra in 1 variable generated by the discriminant b² − ac. (Schur 1968, II.8)
3. The algebra of invariants of the cubic form ax³ + 3bx²y + 3cxy² + dy³ is a polynomial algebra in 1 variable generated by the discriminant 3b²c² + 6abcd − 4b³d − 4c³a − a²d² of degree 4. (Schur 1968, II.8)
4. The algebra of invariants of a quartic form is generated by invariants of degrees 2, 3. (Schur 1968, II.8)
5. The algebra of invariants of a quintic form 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. (Schur 1968, II.9)
6. 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)
7. von Gall (1888) harvtxt error: no target: CITEREFvon_Gall1888 (help) and Dixmier & Lazard (1986) harvtxt error: no target: CITEREFDixmierLazard1986 (help) 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
8. von Gall (1880) harvtxt error: no target: CITEREFvon_Gall1880 (help) 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.
9. Brouwer & Popoviciu (2010a) showed that the algebra of invariants of a degree 9 form is generated by 92 invariants
10. Brouwer & Popoviciu (2010b) showed that the algebra of invariants of a degree 10 form is generated by 106 invariants

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 0961375
• 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: 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
• 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 0722856
• Schur, Issai (1968), Grunsky, Helmut (ed.), Vorlesungen über Invariantentheorie, Die Grundlehren der mathematischen Wissenschaften, vol. 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: 1022–1046, ISSN 0002-9327, MR 0220738
• 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