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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 

 

Double-exponential lower bound for the degree of any system of generators of a polynomial prime ideal


Author: A. L. Chistov
Translated by: The author
Original publication: Algebra i Analiz, tom 20 (2008), nomer 6.
Journal: St. Petersburg Math. J. 20 (2009), 983-1001
MSC (2000): Primary 13P10, 14Q20
DOI: https://doi.org/10.1090/S1061-0022-09-01081-4
Published electronically: October 2, 2009
MathSciNet review: 2530898
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Abstract: Let $ A$ be a polynomial ring in $ n+1$ variables over an arbitrary infinite field $ k$. It is proved that for all sufficiently large $ n$ and $ d$ there is a homogeneous prime ideal $ {\mathfrak{p}}\subset A$ satisfying the following conditions. The ideal $ {\mathfrak{p}}$ corresponds to a component, defined over $ k$ and irreducible over $ \widebar{k}$, of a projective algebraic variety given by a system of homogeneous polynomial equations with polynomials in $ A$ of degrees less than $ d$. Any system of generators of $ {\mathfrak{p}}$ contains a polynomial of degree at least $ d^{2^{cn}}$ for an absolute constant $ c>0$, which can be computed efficiently. This solves an important old problem in effective algebraic geometry. For the case of finite fields a slightly less strong result is obtained.


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Additional Information

A. L. Chistov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia
Email: alch@pdmi.ras.ru

DOI: https://doi.org/10.1090/S1061-0022-09-01081-4
Keywords: Polynomial ideal, projective algebraic variety, Gr\"obner basis, effective algebraic geometry.
Received by editor(s): April 10, 2008
Published electronically: October 2, 2009
Article copyright: © Copyright 2009 American Mathematical Society