Invariant of a binary form: Difference between revisions

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===Covariants of a binary septic===

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. {{harvtxt|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 {{harvtxt|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. {{harvtxt|von Gall|1888}} and {{harvtxt|Dixmier|Lazard|19861988}} 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. {{harvtxt|Cröni|2002}} gives 147 generators for the ring of covariants.

===Covariants of a binary octavic===

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===Covariants of several linear and quadratic forms===

The ring of invariants of a sum of ''m'' linear forms and ''n'' quadratic forms is generated by ''m''(''m''–1)/2 + ''n''(''n''+1)/2 generators in degree 2, ''nm''(''m''+1)/2 + ''n''(''n''–1)(''n''–2)/6 in degree 3, and ''m''(''m''+1)''n''(''n''–1)/4 in degree 4.

is generated by ''m''(''m''–1)/2 + ''n''(''n''+1)/2 generators in degree 2, ''nm

''(''m''+1)/2 + ''n''(''n''–1)(''n''–2)/6 in degree 3, and m''(''m''+1)''n''(''n

''–1)/4 in degree 4.

For the number of generators of the ring of covariants, change ''m'' to ''m''+1.

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*{{Citation | last1=Brouwer | first1=Andries E. | last2=Popoviciu | first2=Mihaela | title=The invariants of the binary nonic | doi=10.1016/j.jsc.2010.03.003 | year=2010a | journal=Journal of Symbolic Computation | issn=0747-7171 | volume=45 | issue=6 | pages=709–720 | mr=2639312| arxiv=1002.0761 | s2cid=30297 }}

*{{Citation | last1=Brouwer | first1=Andries E. | last2=Popoviciu | first2=Mihaela | title=The invariants of the binary decimic | doi=10.1016/j.jsc.2010.03.002 | year=2010b | journal=Journal of Symbolic Computation | issn=0747-7171 | volume=45 | issue=8 | pages=837–843 | mr=2657667| arxiv=1002.1008 | s2cid=12702092 }}

*{{Citation | first=Holger |last=Cröni |title=Zur Berechnung von Kovarianten von Quantiken |type=Dissertation |publisher=Univ. des Saarlandes |___location=Saarbrücken |year=2002}}

*{{Citation | last1=Dixmier | first1=Jacques | last2=Lazard | first2=D. | title=Minimum number of fundamental invariants for the binary form of degree 7 | doi=10.1016/S0747-7171(88)80026-9 | year=1988 | journal=Journal of Symbolic Computation | issn=0747-7171 | volume=6 | issue=1 | pages=113–115 | mr=961375| doi-access=free }}

*{{Citation | last1=von Gall | first1=August Freiherr | title=Das vollständige Formensystem einer binären Form achter Ordnung | doi=10.1007/BF01444117 | year=1880 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=17 | issue=1 | pages=31–51 | mr=1510048| s2cid=120828980 | url=https://zenodo.org/record/2373565 }}