Lower bounds for a polynomial in terms of its coefficients

We determine new sufficient conditions in terms of the coefficients for a polynomial f is an element of R [(X) under bar] of degree 2d (d >= 1) in n >= 1 variables to be a sum of squares of polynomials, thereby strengthening results of Fidalgo and Kovacec (Math. Zeitschrift, to appear) and of Lasserre (Arch. Math. 89 (2007) 390-398). Exploiting these results, we determine, for any polynomial f is an element of R[(X) under bar] of degree 2d whose highest degree term is an interior point in the cone of sums of squares of forms of degree d, a real number r such that f - r is a sum of squares of polynomials. The existence of such a number r was proved earlier by Marshall (Canad. J. Math. 61 (2009) 205-221), but no estimates for r were given. We also determine a lower bound for any polynomial f whose highest degree term is positive definite.