Algebraic sets of types A, B, and C coincide

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.