Geodesic
A Generalization of the Hilbert Basis Theorem
Matematičeskie zametki, Tome 74 (2003) no. 4, pp. 508-516

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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 $a$. 
@article{MZM_2003_74_4_a3,
     author = {K. Yu. Gorbunov},
     title = {A {Generalization} of the {Hilbert} {Basis} {Theorem}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {508--516},
     publisher = {mathdoc},
     volume = {74},
     number = {4},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2003_74_4_a3/}
}
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K. Yu. Gorbunov. A Generalization of the Hilbert Basis Theorem. Matematičeskie zametki, Tome 74 (2003) no. 4, pp. 508-516. http://geodesic.mathdoc.fr/item/MZM_2003_74_4_a3/