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.