There's a useful mathematical fact here, which is that a multivariate polynomial with rational coefficients is nonnegative over the reals if and only if it can be written as a sum of squares of rational functions (see https://en.wikipedia.org/wiki/Hilbert%27s_seventeenth_problem). Rational is important here, as there are positive polynomials that are not the sum of squares of polynomials (there's an example of such on the Wikipedia page).
What I don't know, and what that Wikipedia page is quiet about, is if there is known a general algorithm to either compute such a sum of squares or prove that none exists (or even the weaker problem of just detecting if one exists or not, which is what we really care about here). Another interesting related read is http://andrescaicedo.wordpress.com/2008/11/11/275-positive-polynomials/. Aaron Meurer On Wed, Sep 3, 2014 at 9:25 PM, Chris Smith <[email protected]> wrote: > It would be interesting to see how the positive definite is known. Page > https://en.wikipedia.org/wiki/Positive-definite_matrix gives an example for > which v.T*M*v gives 2*a**2 - 2*a*b + 2*b**2 - 2*b*c + 2*c**2 which is > factored into a sum of squares (and hence nonnegative). I don't know if we > have a way of factoring such expressions into squares, however. > > -- > You received this message because you are subscribed to the Google Groups > "sympy" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/sympy. > To view this discussion on the web visit > https://groups.google.com/d/msgid/sympy/fe87cb3a-721b-4ff6-a188-368ba56f0a9f%40googlegroups.com. > > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/sympy. To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/CAKgW%3D6KougVK5SKxebbYCjQLvcs3UMA9cxgSnOC-6QG_KWwOHA%40mail.gmail.com. For more options, visit https://groups.google.com/d/optout.
