LOWER BOUNDS FOR A POLYNOMIAL IN TERMS OF ITS COEFFICIENTS MEHDI GHASEMI AND MURRAY MARSHALL Abstract. We determine new sufficient conditions in terms of the coefficients for a polynomial f ∈ R[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 [2] and of Lasserre [6]. Exploiting these results, we determine, for any polynomial f ∈ R[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 [8], but no estimates for r were given. We also determine a lower bound for any polynomial f whose highest degree term is positive definite. 1. Introduction Fix a non-constant polynomial f ∈ R[X ]= R[X 1 , ··· ,X n ], where n ≥ 1 is an integer number, and define f ∗ = inf {f (a ) | a ∈ R n }. Let deg(f )= m and decompose f as f = f 0 + ··· + f m (the homogeneous decom- position of f ), where f i is a form of degree i, i =0,...,m. A necessary condition for f ∗ = −∞ is that f m is positive semidefinite (so, in particular, m is even). A sufficient condition for f ∗ = −∞ is that f m is positive definite. We assume from now on that deg(f )=2d, d ≥ 1, i.e., m =2d. For convenience we denote the cone of all sos 1 polynomials by ∑ R[X ] 2 and the cone of all positive semidefinite forms and sos forms of degree 2d by P 2d,n and Σ 2d,n . Also, by R[X ] k we mean the subspace of R[X ] consisting of all polynomials with degree at most k. Define (1) f sos = sup{r ∈ R | f − r ∈  R[X ] 2 }. One can prove that f sos ≤ f ∗ and (2) f sos = inf {ℓ(f ) | ℓ ∈ χ 2d }, where χ 2d is the set of all linear maps ℓ : R[X ] 2d → R such that ℓ(1) = 1 and ℓ(p 2 ) ≥ 0 for all p ∈ R[X ] of degree ≤ d. Computing f ∗ is difficult in general, and one of the successful approaches is to compute f sos instead. This is accomplished by using semidefinite programming (SDP) which is a polynomial time algorithm [5] [9]. The equivalent definitions (1) and (2) for f sos can be considered as two SDP problems which are dual of each other (and the duality gap is 0 in this case). Date : February 1, 2010. 2000 Mathematics Subject Classification. Primary 12D99 Secondary 14P99, 90C22. Key words and phrases. Positive polynomials, sums of squares, optimization. 1 Abbreviation for sum of squares 1