Complexity Estimates for Representations of Schmüdgen Type

Let F={f1,..., fj} and let K be a closed basic set in Rn given by the polynomial inequalities f1 \ 0,..., fj \ 0. Let S{F} be the semiring generated by the fk and the squares in R[x1,..., xn]. For example, if F={f1} then S{F}=s1+s2f1, where s1, s2 are sums of squares of polynomials. Schmüdgen has shown that if K is compact then any polynomial strictly positive on K belongs to S{F}. This paper develops a result of Schmüdgen type for functions in one dimension merely non-negative on K. For this, it is necessary to add additional hypotheses, such as the proximity of complex zeros, to compensate for the loss of strict positivity necessary for Schmüdgen’s result. © 2001 Elsevier Science (USA) 1