Invariant of a binary form: Difference between revisions

Revision as of 02:39, 26 August 2024

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Multiple forms:

* Covariants of several linear forms: The ring of invariants of ''<math>n''</math> linear forms is generated by ''<math>n''(''n''–1-1)/2</math> invariants of degree 2. The ring of covariants of ''<math>n''</math> linear forms is essentially the same as the ring of invariants of ''<math>n''+1</math> linear forms.

* Covariants of several linear forms: The ring of invariants of ''n'' linear forms is generated by ''n''(''n''–1)/2 invariants of degree 2. The ring of covariants of ''n'' linear forms is essentially the same as the ring of invariants of ''n''+1 linear forms.

* Covariants of several linear and quadratic forms:

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

** For the number of generators of the ring of covariants, change ''<math>m''</math> to ''<math>m''+1</math>.

* Covariants of many cubics or quartics: See {{harvtxt|Young|1898}}.

==See also==