On the moment problem of closed semi-algebraic sets

Abstract

The moment problem for compact semi-algebraic sets has been solved in [S1] (see [PD], Section 6.4, for a refinement of the result). In the terminology explained below, this means that each defining sequence f of a compact semialgebraic set Kf has property (SMP ). On the other hand, for many noncompact semi-algebraic sets (for instance, sets containing a cone of dimension two [KM], [PS]) the moment problem is not solvable. Only very few noncompact semi-algebraic sets (classes of real algebraic curves [So], [KM], [PS] and cylinder sets [Mc]) are known to have a positive solution of the moment problem. In this paper we study semi-algebraic setsKf such that there exist polynomials h1, . . ., hn which are bounded on the set Kf . Our main result (Theorem 1) reduces the moment problem for the set Kf to the moment problem for the “fiber sets” Kf ∩Cλ, where Cλ is real algebraic variety {x ∈ R : h1(x) = λ1, . . ., hn(x) = λn}. From this theorem new classes of non-compact closed semi-algebraic sets are obtained for which the moment problem has an affirmative solution. Combined with a result of V. Powers and C. Scheiderer [PS], it follows that tube sets around certain real algebraic curves have property (SMP ) (see Theorem 9). Let f = (f1, . . ., fk) be a finite set of polynomials fj ∈ R[x] ≡ R[x1, . . ., xd] and let K ≡ Kf be the associated closed semi-algebraic subset defined by