Invariant of a binary form: Difference between revisions

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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"

|+ (basic invariant, basic covariant)

! Degree of forms !! 1 !! 2 !! 3 !! 4 !! 5

!n

|-

| 1

| (1, 3)

|| (2, 5)

|| (4, 13)

|| (5, 20)

|| (23, 94)

|

|-

| 2

|

|| (3, 6)

|| (5, 15)

|| (6, 18)

|| (29, 92)

|

|-

| 3

|

||

||

|| (20, 63)

||

|

|-

| 4

|

||

||

|| (8, 28)

||

|

|}

Notes:

===Covariants of two linear forms===

There are 1 basic invariant and 3 basic covariants.

===Covariants of a linear form and a quadratic===

There are 2 basic invariants and 5 basic covariants.

===Covariants of a linear form and a cubic===

There are 4 basic invariants (essentially the covariants of a cubic) and 13 basic covariants.

===Covariants of a linear form and a quartic===

There are 5 basic invariants (essentially the basic covariants of a quartic) and 20 basic covariants.

===Covariants of a linear form and a quintic===

There are 23 basic invariants (essentially the basic covariants of a quintic) and 94 basic covariants.

===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 two quadratics===

There are 3 basic invariants and 6 basic covariants.

===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.

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

===Covariants of a quadratic and a cubic===

There are 5 basic invariants and 15 basic covariants

===Covariants of a quadratic and a quartic===

There are 6 basic invariants and 18 basic covariants

===Covariants of a quadratic and a quintic===

There are 29 basic invariants and 92 basic covariants

===Covariants of a cubic and a quartic===

There are 20 basic invariants and 63 basic covariants

* The basic invariants of a linear form are essentially the same as its basic covariants.

===Covariants of two quartics===

* For two quartics, 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.)

Multiple forms:

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 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 many cubics or quartics===

* 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.

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

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

See {{harvtxt|Young|1898}}.

==See also==