The core variety of a multisequence in the truncated moment problem.

Let β ≡ β ( m ) = { β i } i ∈ Z + n , | i | ≤ m , β 0 > 0 , denote a real n -dimensional multisequence of finite degree m . The Truncated Moment Problem concerns the existence of a positive Borel measure μ , supported in R n , such that (0.1) β i = ∫ R n x i d μ ( i ∈ Z + n , | i | ≤ m ) . We associate to β ≡ β ( 2 d ) an algebraic variety in R n called the core variety , V ≡ V ( β ) . The core variety contains the support of each representing measure μ . We show that if V is nonempty, then β ( 2 d − 1 ) has a representing measure. Moreover, if V is a nonempty compact or determining set, then β ( 2 d ) has a representing measure. We also use the core variety to exhibit a sequence β , with positive definite moment matrix and positive Riesz functional, which fails to have a representing measure. [ABSTRACT FROM AUTHOR]