Lower bounds for a polynomial in terms of its coefficients

We determine new sufficient conditions in terms of the coefficients for a polynomial ${f\in \mathbb{R}\underline{X}]}We determine new sufficient conditions in terms of the coefficients for a polynomial f ? \mathbbRX]{f\in \mathbb{R}\underline{X}]} 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 ? \mathbbRX]{f\in \mathbb{R}\underline{X}]} 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.