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Each univariate complex polynomial has a ‘big’ factor

Published online by Cambridge University Press:  14 November 2011

Ph. Glesser ,
M. Mignotte  and
M. Petkovic
Ph. Glesser
Affiliation:
U.F.R. de mathématique et d'informatique, Université Louis Pasteur, 7 rue René Descartes, F-67084 Strasbourg Cedex, France
M. Mignotte
Affiliation:
U.F.R. de mathématique et d'informatique, Université Louis Pasteur, 7 rue René Descartes, F-67084 Strasbourg Cedex, France
M. Petkovic
Affiliation:
U.F.R. de mathématique et d'informatique, Université Louis Pasteur, 7 rue René Descartes, F-67084 Strasbourg Cedex, France
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Abstract

We consider the following problem: Let P be a monic polynomial of degree n with complex coefficients. What can be the maximum ‘size’ of a monic divisor Q of P? Here the size of a polynomial R is the maximum ||R|| of the moduli of its values on the unit circle. In 1991, B. Beauzamy proved that there exists a divisor Q with ||Q|| ≧ e∈n−1, ∈ = 0.0019, when all the roots of P belong to the unit circle. Using a very recent result of D. Boyd, we obtain a general result which, in the same case, gives ||Q||≧βn; here β = 1.38135 … is optimal.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 1 , 1994 , pp. 71 - 76
Copyright
Copyright © Royal Society of Edinburgh 1994

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References

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