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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Polynomials with roots modulo every integer


Authors: Daniel Berend and Yuri Bilu
Journal: Proc. Amer. Math. Soc. 124 (1996), 1663-1671
MSC (1991): Primary 11R09, 11R45; Secondary 11D61, 11U05
MathSciNet review: 1307495
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Abstract: Given a polynomial with integer coefficients, we calculate the density of the set of primes modulo which the polynomial has a root. We also give a simple criterion to decide whether or not the polynomial has a root modulo every non-zero integer.


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Additional Information

Daniel Berend
Affiliation: Department of Mathematics and Computer Science, Ben-Gurion University, Beer Sheva 84105, Israel
Email: berend@black.bgu.ac.il

Yuri Bilu
Affiliation: Department of Mathematics and Computer Science, Ben-Gurion University, Beer Sheva 84105, Israel and Université Bordeaux 2, Mathématiques Stochastiques, BP26, F-33076 Bordeaux Cedex, France
Address at time of publication: Max Planck Institute for Mathematics, Gottfried Claren Str. 26, 53225 Bonn, Germany
Email: yuri@cfgauss.uni-math.gwdg.de

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03210-8
PII: S 0002-9939(96)03210-8
Keywords: Diophantine equations, congruences, effective number theory, Poincar\'{e} sets
Received by editor(s): March 7, 1994
Received by editor(s) in revised form: November 28, 1994
Communicated by: William W. Adams
Article copyright: © Copyright 1996 American Mathematical Society