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I am a student of Saint Petersburg State Polytechnical University, chair of Theoretical Mechanics. While looking into stability of ideal crystal lattices (2D and 3D) by means of molecular dynamics I have encountered a challenging analytical problem. To draw the stability regions I need to find the conditions on coefficients of certain homogeneous polynomials. Thus, three questions have occurred:

1) When is a homogeneous polynomial of the third degree in three variables over R positive on the positive octant?

2) When is a quadratic form in three variables over R positive on the positive octant?

3) When is a homogeneous polynomial of fourth degree in two variables over R positive?

Currently, I managed to write down only sufficient conditions which all actually base on what can be said about a quadratic form on the positive quadrant.

I would be grateful for any ideas on how to solve such problems. I apologize for any mistakes I might have made as far as terms are concerned.

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Do you actually care about whether the polynomials are strictly positive, or just nonnegative? This may affect the answer, and if you mean strict positivity then you have to say "except at the origin" in the case of the homogeneous polynomials, because these will always be zero at the origin. Or perhaps you mean strict positivity on the strict positive orthant? – Noah Stein Nov 2 '10 at 19:42
As to questions 1) and 3), strict positivity is required. But in question 2) the quadratic form should be non-negative. – Kate Podolskaya Nov 10 '10 at 15:38
up vote 1 down vote accepted

Ms. Podolskaya,

Your 2nd question is related to matrix copositivity, I believe. Take a look at the 5th chapter of Parrilo's doctoral dissertation.

A quadratic form in $\mathbb{R}[x_1,x_2,x_3]$ is of the form

$$P (x_1,x_2,x_3) = \left[\begin{array}{c} x_1\\\ x_2\\\ x_3\end{array}\right]^T \left[\begin{array}{ccc} q_{11} & q_{12} & q_{13}\\\ q_{12} & q_{22} & q_{23}\\\ q_{13} & q_{23} & q_{33}\end{array}\right] \left[\begin{array}{c} x_1\\\ x_2\\\ x_3\end{array}\right]$$

or, more compactly, $P (x) = x^T Q x$. You ask: when is $P$ positive on the positive octant? If $P > 0$ when $x > 0$, then

$$(\forall x \in \mathbb{R^3}) (x > 0 \implies x^T Q x > 0)$$

and, in theory, one could use quantifier elimination to obtain conditions on the $q_{ij}$ coefficients so that $P > 0$ on the positive octant. The following REDLOG script

% positivity on the positive octant

load_package redlog;
rlset ofsf;

% define quadratic form 
P := 1 * q11 * x1 * x1 +
   + 1 * q22 * x2 * x2 +
   + 1 * q33 * x3 * x3 +
   + 2 * q12 * x1 * x2 +
   + 2 * q13 * x1 * x3 +
   + 2 * q23 * x2 * x3;

% define universally quantified formula
phi := all({x1,x2,x3}, (x1 > 0 and x2 > 0 and x3 > 0) impl P>0);

% perform quantifier elimination
rlqe phi;


produces results in a few seconds, but the conditions on the $q_{ij}$ coefficients are enormously long quantifier-free formulas. So enormous that REDUCE crashed!

If $P$ is nonnegative on the nonnegative octant, then

$$(\forall x \in \mathbb{R^3}) (x \geq 0 \implies x^T Q x \geq 0)$$

which is equivalent to saying that matrix $Q = Q^T$ is copositive.

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There is an old theorem by George Pólya which says the following:

Theorem: If $p \in \mathbb R[x_1,\dots,x_n]$ is a homogenous polynomial which is positive on the positive "octant", then for large $k$, the coefficients of the polynomial $(x_1 + \cdots +x_n)^k \cdot p(x_1,\dots,x_n)$ are positive.

Note that the condition in the theorem is necessary and sufficient since positivity of the coefficients of $(x_1 + \cdots +x_n)^k \cdot p(x_1,\dots,x_n)$ also implies that $p$ was positive on the positive "octant".

For a reference or even a proof of this theorem you can look here. These are slides for a talk by Mari Castle called "Everything you’ve ever wanted to know about Pólya’s Theorem (but were afraid to ask)."

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Thank you very much! – Kate Podolskaya Nov 12 '10 at 9:38

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