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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

Martin Sleziak
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Giorgio Busoni
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1 Answers1

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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.

Rodrigo de Azevedo
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  • Thanks. I was able to understand that homogeneous quartic forms can always be written in that way but I was missing the last step that the matrix M is not unique but there is a whole subspace of them. So If I consider the whole subspace I can see if it contains any semidefinite positive matrix. If yes, then as there is a SOS decomposition global positivity is guaranteed. AnywayReading theorem 3.2 I see that if m=4 and n>3 is not guaranteed that a positive quartic form has an SOS, so this method would fail in that case, right? – Giorgio Busoni Jul 19 '16 at 14:30
  • @GiorgioBusoni It's a sufficient condition. If a form has an SOS decomposition, then we conclude it is globally nonnegative. If the SDP is infeasible, then there is no SOS decomposition and we cannot conclude anything regarding global nonnegativity. – Rodrigo de Azevedo Jul 19 '16 at 14:38