Conditions on coefficients of a homogeneous polynomial of the third degree in three variables over R which allow it to be positive on a positive octant

Question

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.

Answer 1

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;

end;

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.

Answer 2

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)."