Representations of non-negative polynomials having finitely many zeros

Let K be a basic closed semialgebraic set in R defined by polynomial inequalities g1 ≥ 0, . . . , gs ≥ 0, where g1, . . . , gs ∈ R[x1, . . . , xn], and let f ∈ R[x1, . . . , xn]. In [12], Schmüdgen proves that, if K is compact and f is strictly positive on K, then f belongs to the quadratic preordering generated by g1, . . . , gs. Denote by M the (smaller) quadratic module generated by g1, . . . , gs. Results of Putinar [8] and Jacobi [1] show that, if M is archimedean, then f > 0 on K implies f ∈ M . The question of exactly when M is archimedean is studied in detail in [2]. In [11], extending earlier results in the preordering case in [10], Scheiderer gives sufficient conditions for f ≥ 0 on K to imply f ∈ M . In [11, Cor. 2.6], as an application of his methods, Scheiderer extends the Putinar-Jacobi result to include the case where f ≥ 0 on K and, at each zero of f in K, the partial derivatives of f vanish and the hessian of f is positive definite. The Putinar-Jacobi result serves as the theoretical underpinning for an optimization algorithm based on semidefinite programming due to Lasserre; see [4] or [5]. According to the Putinar-Jacobi result, if M is archimedean, the minimum value of any polynomial f on K is equal to sup{c ∈ R | f − c ∈ M}. This latter number can be approximated by Lasserre’s algorithm. One is naturally interested in knowing when f − c ∈ M holds when c is the exact minimum of f on K, e.g., see [4, Th. 2.1 and Remark 2.2]. Although [11, Cor. 2.6] sheds light on this question, its usefulness is limited by the unrealistic constraints on the boundary zeros. In Section 1 we review basic terminology and results and, at the same time, we use the Basic Lemma in [3] to give a short proof of the main result in [11]. In Section 2 we prove that the constraints on the boundary zeros in [11, Cor. 2.6] can be replaced by constraints which are much less restrictive and much more natural; see Theorem 2.3. In the Appendix, we examine the Basic Lemma in [3], and we compare this result to Lemma 2.6 in [10], which is the key result in the approach taken by Scheiderer in [10] and [11].