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A generalization of the Hilbert basis theorem. (English. Russian original) Zbl 1109.12001

Math. Notes 74, No. 4, 483-490 (2003); translation from Mat. Zametki 74, No. 4, 508-516 (2003).
Summary: 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 \(a\).

MSC:

12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
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