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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

   

 

A bound for the degree of a system of equations determining the variety of reducible polynomials


Author: A. L. Chistov
Translated by: the author
Original publication: Algebra i Analiz, tom 24 (2012), nomer 3.
Journal: St. Petersburg Math. J. 24 (2013), 513-528
MSC (2010): Primary 14Q15; Secondary 14M99, 12Y05, 12E05, 13P05
Published electronically: March 21, 2013
MathSciNet review: 3014133
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {\mathbb{A}}^N(\bar {K})$ denote the affine space of homogeneous polynomials of degree $ d$ in $ n+1$ variables with coefficients from the algebraic closure $ \bar {K}$ of a field $ K$ of arbitrary characteristic; so $ N=\binom {n+d}{n}$. It is proved that the variety of all reducible polynomials in this affine space can be given by a system of polynomial equations of degree less than $ 56d^7$ in $ N$ variables. This result makes it possible to formulate an efficient version of the first Bertini theorem for the case of a hypersurface.


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Additional Information

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

DOI: http://dx.doi.org/10.1090/S1061-0022-2013-01251-9
Keywords: Absolute irreducibility, lattices, Bertini theorem
Received by editor(s): November 1, 2011
Published electronically: March 21, 2013
Article copyright: © Copyright 2013 American Mathematical Society