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Ternary quartic

Ternary quartic

In mathematics, a ternary quartic form is a degree 4 homogeneous polynomial in three variables.

Hilbert's theorem

showed that a positive semi-definite ternary quartic form over the reals can be written as a sum of three squares of quadratic forms.

Invariant theory

1908}} on invariant theory. This table collects 202 of the 331 invariants of ternary biquadratic forms. These forms are graded in two variables ''x'' and ''u''. The horizontal direction of the table lists the invariants with increasing grades in ''x'', while the vertical direction lists them with increasing grades in ''u''.

The ring of invariants is generated by 7 algebraically independent invariants of degrees 3, 6, 9, 12, 15, 18, 27 (discriminant) , together with 6 more invariants of degrees 9, 12, 15, 18, 21, 21, as conjectured by . discussed the invariants of order up to about 15.

The Salmon invariant is a degree 60 invariant vanishing on ternary quartics with an inflection bitangent.

Catalecticant

The catalecticant of a ternary quartic is the resultant of its 6 second partial derivatives. It vanishes when the ternary quartic can be written as a sum of five 4th powers of linear forms.

References

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This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

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