Invariant of a binary form: Difference between revisions

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==The ring of invariants==

The structure of the ring of invariants has been worked out for small degrees. as{{harvtxt|Sylvester|Franklin|1879}} follows:gave tables of the numbers of generators of invariants and covariants for forms of degree up to 10.

# TheFor linear forms ''ax'' + ''by'' the only invariants are constants.

# The algebra of invariants of the binaryquadratic form ''ax''² + 2''bxy'' + ''cy''² is a polynomial algebra in 1 variable generated by the discriminant ''b''² − ''ac''. {{harv|Schur|1968|loc=II.8}}

# The algebra of invariants of the ternary form ''ax''³ + 3''bx''²''y'' + 3''cxy''² + ''dy''³ is a polynomial algebra in 1 variable generated by the discriminant 3''b''²''c''² + 6''abcd'' − 4''b''³''d'' − 4''c''³''a'' − ''a''²''d''² of degree 4. {{harv|Schur|1968|loc=II.8}}

# The algebra of invariants of a quaternary form is generated by invariants of degrees 2, 3. {{harv|Schur|1968|loc=II.8}}

# The algebra of invariants of a quintic form is generated by invariants of degree 4, 8, 12, 18. The generators of degrees 4, 8, 12 generate a polynomial ring, which contains the square of the generator of degree 18. {{harv|Schur|1968|loc=II.9}}

# The algebra of invariants of a sextic form is generated by invariants of degree 2, 4, 6, 10, 15. The generators of degrees 2, 4, 6, 10 generate a polynomial ring, which contains the square of the generator of degree 15. {{harv|Schur|1968|loc=II.9}}

#{{harvtxt|von Gall|1888}} and {{harvtxt|Dixmier|Lazard|1986}} Theshowed that the algebra of invariants of a degree 7 form is generated by a set with 1 invariant of degree 4, 3 of degree 8, 6 of degree 12, 4 of degree 14, 2 of degree 16, 9 of degree 18, and one of each of the degrees 20, 22, 26, 30

#{{harvtxt|von Gall|1880}}, and {{harvtxt|Shioda|1967}} Theshowed that the algebra of invariants of a degree 8 form is generated by 9 invariants of degrees 2, 3, 4, 5, 6, 7, 8, 9, 10, and the ideal of relations between them is generated by elements of degrees 16, 17, 18, 19, 20.

#{{harvtxt|Brouwer|Popoviciu|2010a}} Generatedshowed that the algebra of invariants of a degree 9 form is generated by 92 invariants

#{{harvtxt|Brouwer|Popoviciu|2010b}} Generatedshowed that the algebra of invariants of a degree 10 form is generated by 106 invariants

==References==