Sufficient conditions for a real polynomial to be a sum of squares

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.