A Criterion for Positive Polynomials

Let F be a homogeneous real polynomial of even degree in any number of variables. We consider the problem of giving explicit conditions on the coefficients so that F is positive definite or positive semi-definite. In this note we produce a necessary condition for positivity and a sufficient condition for non-negativity, in terms of positivity or semi-positivity of a one-variable characteristic polynomial of F . Also, we review another well-known sufficient condition. §1 (1.1) Let F be a homogeneous polynomial of degree d in n variables x1, . . . , xn with coefficients in a field K. We will denote K(n, d) the K-vector space of all such polynomials. Its dimension is N = ( n−1+d d ) . For K = R, the field of real numbers, we shall say that F is positive (resp. non-negative), written F > 0 (resp. F ≥ 0), if F (x) > 0 for all x ∈ R − {0} (resp. F (x) ≥ 0 for all x ∈ R). 1991 Mathematics Subject Classification. 14Pxx, 14Mxx.