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:
- The only invariants are constants.
- The ring of invariants is a polynomial ring in 1 variable generated by the discriminant
- Classical
- Classical
- Classical
- Classical
- von Gall (1888) Dixmier & Lazard (1986)
- von Gall (1880), Shioda (1967) The ring 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.
- Brouwer & Popoviciu (2009)
- Brouwer & Popoviciu (2010) 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 7terOrdnung", Mathematische Annalen, 31 (3): 318–336, doi:10.1007/BF01206218, ISSN 0025-5831, MR 1510486
- Hilbert, David (1993) [1897], Theory of algebraic invariants, Cambridge University Press, ISBN 978-0-521-44457-6, MR 1266168
- 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