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{{main|Glossary of invariant theory}}
A binary form (of degree ''n'') is a homogeneous polynomial
In terms of [[representation theory]], given any representation
The invariants of a binary form form a [[graded algebra]], and {{harvtxt|Gordan|1868}} proved that this algebra is finitely generated if the base field is the complex numbers.
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===Covariants of a binary linear form===
For linear forms
===Covariants of a binary quadric===
The algebra of invariants of the quadratic form
===Covariants of a binary cubic===
The algebra of invariants of the cubic form
===Covariants of a binary quartic===
The algebra of invariants of a quartic form is generated by invariants
<math display="block"> \begin{aligned} F_4(x, y)&=A x^4+4 B x^3 y+6 C x^2 y^2+4 D x y^3+E y^4\\ i_{F_4}&=A E-4 B D+3 C^2 \\ j_{F_4}&=A C E+2 B C D-C^3-B^2 E-A D^2 \end{aligned} </math> This ring is naturally isomorphic to the ring of modular forms of level 1, with the two generators corresponding to the Eisenstein series ===Covariants of a binary quintic===
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==Invariants of several binary forms==
The covariants of a binary form are essentially the same as joint invariants of a binary form and a binary linear form. More generally, on can ask for the joint invariants (and covariants) of any collection of binary forms. Some cases that have been studied are listed below.
{| class="wikitable"
! Degree of forms !! 1 !! 2 !! 3 !! 4 !! 5
|-
| 1
| (1, 3)
|| (2, 5)
|| (4, 13)
|| (5, 20)
|| (23, 94)
|-
| 2
|
|| (3, 6)
|| (5, 15)
|| (6, 18)
|| (29, 92)
|-
| 3
|
||
||
|| (20, 63)
||(8, 28)
|-
| 4
|
||
||
||
||
|}
Notes:
===Covariants of two linear forms===▼
▲There are 1 basic invariant and 3 basic covariants.
There are 4 basic invariants (essentially the covariants of a cubic) and 13 basic covariants.▼
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.▼
For the number of generators of the ring of covariants, change ''m'' to ''m''+1.▼
There are 8 basic invariants (3 of degree 2, 4 of degree 3, and 1 of degree 4) and 28 basic covariants. (Gordan gave 30 covariants, but Sylvester showed that two of these are reducible.)▼
▲
===Covariants of many cubics or quartics===▼
▲
Multiple forms:
* Covariants of several linear forms: The ring of invariants of <math>n</math> linear forms is generated by <math>n(n-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.
▲** The ring of invariants of a sum of
▲** For the number of generators of the ring of covariants, change
==See also==
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