Invariant of a binary form: Difference between revisions

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===Covariants of a binary linear form===

For linear forms <math>F_1(x,y) = Ax + By</math> the only invariants are constants. The algebra of covariants is generated by the form itself of degree 1 and order 1.

===Covariants of a binary quadric===

The algebra of invariants of the quadratic form <math>F_2(x,y) = Ax^2 + 2Bxy + Cy^2</math> is a polynomial algebra in 1 variable generated by the discriminant <math>B^2 - AC</math> of degree 2. The algebra of covariants is a polynomial algebra in 2 variables generated by the discriminant together with the form ''F'' itself (of degree 1 and order 2). {{harv|Schur|1968|loc=II.8}} {{harv|Hilbert|1993|loc=XVI, XX}}

===Covariants of a binary cubic===

The algebra of invariants of the cubic form <math>F_3(x,y) = Ax^3 + 3Bx^2y + 3Cxy^2 + Dy^3</math> is a polynomial algebra in 1 variable generated by the discriminant <math>\Delta = 3B^2C^2 + 6ABCD- 4B^3D - 4C^3A - A^2D^2</math> of degree 4. The algebra of covariants is generated by the discriminant, the form itself (degree 1, order 3), the Hessian <math>H</math> (degree 2, order 2) and a covariant <math>T</math> of degree 3 and order 3. They are related by the [[Syzygy (mathematics)|syzygy]] <math>4H^3=Df^2-T^2</math> of degree 6 and order 6. {{harv|Schur|1968|loc=II.8}} {{harv|Hilbert|1993|loc=XVII, XX}}

===Covariants of a binary quartic===

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|+ (basic invariant, basic covariant)

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

!n

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| 1

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| 2

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| 3

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|| (20, 63)

||(8, 28)

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| 4

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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==