A Generalization of the Hilbert Basis Theorem

A generalization of the Hilbert basis theorem in the geometric setting is proposed. It asserts that, for any well-describable (in a certain sense) family of polynomials, there exists a number $C$ such that if $P$ is an everywhere dense (in a certain sense) subfamily of this family, $a$ is an arbitrary point, and the first $C$ polynomials in any sequence from $P$ vanish at the point $a$, then all polynomials from $P$ vanish at $a$.