  The question how to certify non-negativity of a polynomial function lies at
the heart of Real Algebra and also has important applications to Optimization.
In this article we investigate the question of non-negativity in the context of
multisymmetric polynomials. In this setting we generalize the characterization
of non-negative symmetric polynomials by adapting the method of proof developed
by the second author. One particular case where our results can be applied is
the question of certifying that a (multi-)symmetric polynomial defines a convex
function. As a direct corollary of our main result we are able to derive that
in the case of (multi-)symmetric polynomials of a fixed degree testing for
convexity can be done in a time which is polynomial in the number of variables.
This is in sharp contrast to the general case, where it is known that testing
for convexity is NP-hard already in the case of quartic polynomials.
