Conditions for a Real Polynomial to Be Sum of Squares

We provide explicit conditions for a 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.