Invariant of a binary form: Difference between revisions

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

# The algebra of invariants of the quadratic 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 ternarycubic 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 quaternaryquartic 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}}