Invariant of a binary form

In mathematics, 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.

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.

The ring of invariants

The structure of the ring of invariants has been worked out for small degrees as follows:

1. The only invariants are constants.
2. The algebra of invariants of the binary form ax² + 2bxy + cy² is a polynomial algebra in 1 variable generated by the discriminant b² − ac
3. The algebra of invariants of the ternary form ax³ + 3bx²y + 3cxy² + dy³ is a polynomial algebra in 1 variable generated by the discriminant 3b²c² + 6abcd − 4b³d − 4c³a − a²d² of degree 4.
4. The algebra of invariants is generated by invariants of degrees 2, 3.
5. The algebra of invariants is generated by invariants of degree 4, 8, 12, 18
6. The algebra of invariants is generated by invariants of degree 2, 4, 6, 10, 15
7. von Gall (1888) harvtxt error: no target: CITEREFvon_Gall1888 (help) Dixmier & Lazard (1986) harvtxt error: no target: CITEREFDixmierLazard1986 (help) The algebra of invariants 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), Shioda (1967) The algebra of invariants 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) Generated by 92 invariants
10. Brouwer & Popoviciu (2010b) 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
• Schur, Issai (1968), Grunsky, Helmut (ed.), Vorlesungen über Invariantentheorie, Die Grundlehren der mathematischen Wissenschaften, Band 143, vol. 143, Berlin, New York: Springer-Verlag, 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