Sufficient conditions for a real polynomial to be a sum of squares
  We provide explicit conditions for a real polynomial $f$ of degree 2d to be a sum of squares (s.o.s.), stated only in terms of the coefficients of $f$, i.e. with no lifting. All conditions are simple and provide an explicit description of a convex polyhedral subcone of the cone of s.o.s. polynomials of degree at most 2d. We also provide a simple condition to ensure that $f$ is s.o.s., possibly modulo a constant. 
Jean B. Lasserre