Algebraic sets of types A, B, and C coincide

TL;DR

This paper proves the equivalence of algebraic sets of types A, B, and C, clarifying their definitions and resolving longstanding questions in the theory of multidimensional moment problems and polynomial orthogonality.

Contribution

It establishes that algebraic sets of types A, B, and C are equivalent, answering multiple open questions from 1992 and 2005.

Findings

Type A sets are independent of polynomial choice.

Type B sets are always of type A.

Type A and C sets are equivalent.

Abstract

It is proved that the definition of an algebraic set of type ${\sf A}$ (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 ${\sf B}$ is always of type ${\sf A}$. 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 ${\sf A}$ if and only if it is of type ${\sf C}$ (a notion linked to orthogonality of polynomials of several variables). This, in turn, enables us to answer three questions posed in 2005 by Cicho\'n, Stochel, and Szafraniec.