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.



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
Ms. Podolskaya,


Your 2nd question is related to [matrix copositivity](http://en.wikipedia.org/wiki/Copositive_matrix), I believe. Take a look at the 5th chapter of [Parrilo's doctoral dissertation](http://resolver.caltech.edu/CaltechETD%3Aetd-05062004-055516).


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](http://en.wikipedia.org/wiki/Quantifier_elimination) to obtain conditions on the $q\_{ij}$ coefficients so that $P > 0$ on the positive octant. The following [REDLOG](http://redlog.dolzmann.de/) 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](http://www.reduce-algebra.com/) 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](http://en.wikipedia.org/wiki/Copositive_matrix).


