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 - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T10:13:25Z http://mathoverflow.net/feeds/question/44580 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/44580/conditions-on-coefficients-of-a-homogeneous-polynomial-of-the-third-degree-in-thr 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 Kate Podolskaya 2010-11-02T16:32:55Z 2012-09-30T23:11:07Z <p>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:</p> <p>1) When is a homogeneous polynomial of the third degree in three variables over R positive on the positive octant?</p> <p>2) When is a quadratic form in three variables over R positive on the positive octant?</p> <p>3) When is a homogeneous polynomial of fourth degree in two variables over R positive?</p> <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/44580/conditions-on-coefficients-of-a-homogeneous-polynomial-of-the-third-degree-in-thr/44595#44595 Answer by Andreas Thom for 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 Andreas Thom 2010-11-02T17:56:24Z 2010-11-02T22:39:52Z <p>There is an old theorem by George Pólya which says the following:</p> <blockquote> <p><strong>Theorem:</strong> 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.</p> </blockquote> <p>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".</p> <p>For a reference or even a proof of this theorem you can look <a href="http://userwww.service.emory.edu/~cmagnan/ACEStalks/aces_polya_talk_2.pdf" rel="nofollow">here</a>. 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)."</p> http://mathoverflow.net/questions/44580/conditions-on-coefficients-of-a-homogeneous-polynomial-of-the-third-degree-in-thr/101038#101038 Answer by Rod Carvalho for 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 Rod Carvalho 2012-07-01T03:16:12Z 2012-09-30T23:11:07Z <p>Ms. Podolskaya,</p> <p>Your 2nd question is related to <a href="http://en.wikipedia.org/wiki/Copositive_matrix" rel="nofollow">matrix copositivity</a>, I believe. Take a look at the 5th chapter of <a href="http://resolver.caltech.edu/CaltechETD%3Aetd-05062004-055516" rel="nofollow">Parrilo's doctoral dissertation</a>.</p> <p>A quadratic form in $\mathbb{R}[x_1,x_2,x_3]$ is of the form</p> <p>$$P (x_1,x_2,x_3) = \left[\begin{array}{c} x_1\\ x_2\\ x_3\end{array}\right]^T \left[\begin{array}{ccc} q_{11} &amp; q_{12} &amp; q_{13}\\ q_{12} &amp; q_{22} &amp; q_{23}\\ q_{13} &amp; q_{23} &amp; q_{33}\end{array}\right] \left[\begin{array}{c} x_1\\ x_2\\ x_3\end{array}\right]$$</p> <p>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</p> <p>$$(\forall x \in \mathbb{R^3}) (x > 0 \implies x^T Q x > 0)$$</p> <p>and, in theory, one could use <a href="http://en.wikipedia.org/wiki/Quantifier_elimination" rel="nofollow">quantifier elimination</a> to obtain conditions on the $q_{ij}$ coefficients so that $P > 0$ on the positive octant. The following <a href="http://redlog.dolzmann.de/" rel="nofollow">REDLOG</a> script</p> <pre><code>% 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 &gt; 0 and x2 &gt; 0 and x3 &gt; 0) impl P&gt;0); % perform quantifier elimination rlqe phi; end; </code></pre> <p>produces results in a few seconds, but the conditions on the $q_{ij}$ coefficients are enormously long quantifier-free formulas. So enormous that <a href="http://www.reduce-algebra.com/" rel="nofollow">REDUCE</a> crashed!</p> <p>If $P$ is nonnegative on the nonnegative octant, then</p> <p>$$(\forall x \in \mathbb{R^3}) (x \geq 0 \implies x^T Q x \geq 0)$$</p> <p>which is equivalent to saying that matrix $Q = Q^T$ is <a href="http://en.wikipedia.org/wiki/Copositive_matrix" rel="nofollow">copositive</a>.</p>