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If I have a quartic form that I can write as

$$P(x,y)=(x^2/2,y^2/2,xy)M(x^2/2,y^2/2,xy)$$

where $M$ a a $n \times n$ symmetric matrix, what is the simplest way to derive whether the form is positive definite?

For quadratic forms this is quite easy, you just need to check the eigenvalues of $M$, but for quartic forms written in this way the condition looks less obvious.

In this example the form depends on only $2$ variables and $n=3$, but I am interested in applying this to forms with many more variables

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1 Answer

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Use semidefinite programming (SDP) to compute the sum of squares (SOS) decomposition of a given quartic form. If an SOS decomposition exists, then the quartic form is globally nonnegative.

From one of Parrilo's papers [0]:

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[0] Pablo Parrilo, Semidefinite programming relaxations for semialgebraic problems, 2001.

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