We provide explicit sufficient conditions for a polynomial $f$ to be a sum of squares (s.o.s.), linear in the coefficients of $f$. 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 after adding a constant.
Report #06789, LAAS, Toulouse, France. To appear in Archiv der Mathematik.