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→The ring of invariants: covariants |
→The ring of invariants: degrees orders 1, 2 |
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The structure of the ring of invariants has been worked out for small degrees. {{harvtxt|Sylvester|Franklin|1879}} gave tables of the numbers of generators of invariants and covariants for forms of degree up to 10.
# For linear forms ''ax'' + ''by'' the only invariants are constants. The algebra of covariants is generated by the form itself of degree 1 and order 1.
# The algebra of invariants of the quadratic form ''ax''<sup>2</sup> + 2''bxy'' + ''cy''<sup>2</sup> is a polynomial algebra in 1 variable generated by the discriminant ''b''<sup>2</sup> − ''ac'' 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}}
# The algebra of invariants of the cubic form ''ax''<sup>3</sup> + 3''bx''<sup>2</sup>''y'' + 3''cxy''<sup>2</sup> + ''dy''<sup>3</sup> is a polynomial algebra in 1 variable generated by the discriminant 3''b''<sup>2</sup>''c''<sup>2</sup> + 6''abcd'' − 4''b''<sup>3</sup>''d'' − 4''c''<sup>3</sup>''a'' − ''a''<sup>2</sup>''d''<sup>2</sup> of degree 4. {{harv|Schur|1968|loc=II.8}}
# The algebra of invariants of a quartic form is generated by invariants ''i'', ''j'' of degrees 2, 3. The algebra of covaraints is generated by these two invaraints together with the form ''f'' of degree 1 and order 4, the Hessian ''H'' of degree 2 and order 4, and a covaraint ''T'' of degree 3 and order 6. They are related by a syzygy ''jf''<sup>3</sup>−''Hf''<sup>2</sup>''i''+4''H''<sup>3</sup>+''T''<sup>2</sup>=0 of degree 6 and order 12. {{harv|Schur|1968|loc=II.8}} {{harv|Hilbert|1993|loc=XVIII, XXII}}
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