Algebraic sets of types A, B, and C coincide

Abstract

It is proved that the definition of an algebraic set of type $\mathsf{A}$ (a notion related to the multidimensional Hamburger moment problem) does not depend on the choice of a polynomial describing the algebraic set in question and that an algebraic set of type $\mathsf{B}$ is always of type $\mathsf{A}$. This answers in the affirmative two questions posed in 1992 by the second author. It is also shown that an algebraic set is of type $\mathsf{A}$ if and only if it is of type $\mathsf{C}$ (a notion linked to orthogonality of polynomials of several variables). This, in turn, enables us to answer three questions posed in 2005 by Cichoń, Stochel, and Szafraniec.

The paper was written in December 2007 and not published anywhere else