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

 

Decomposition of polynomials and approximate roots


Author: Arnaud Bodin
Journal: Proc. Amer. Math. Soc. 138 (2010), 1989-1994
MSC (2010): Primary 13B25
Published electronically: February 2, 2010
MathSciNet review: 2596034
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Abstract: We state a kind of Euclidian division theorem: given a polynomial $ P(x)$ and a divisor $ d$ of the degree of $ P$, there exist polynomials $ h(x),Q(x),R(x)$ such that $ P(x) = h\circ Q(x) +R(x)$, with $ \deg h=d$. Under some conditions $ h,Q,R$ are unique, and $ Q$ is the approximate $ d$-root of $ P$. Moreover we give an algorithm to compute such a decomposition. We apply these results to decide whether a polynomial in one or several variables is decomposable or not.


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Additional Information

Arnaud Bodin
Affiliation: Laboratoire Paul Painlevé, Mathématiques, Université de Lille 1, 59655 Villeneuve d’Ascq, France
Email: Arnaud.Bodin@math.univ-lille1.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10245-7
PII: S 0002-9939(10)10245-7
Keywords: Decomposable and indecomposable polynomials in one or several variables
Received by editor(s): March 10, 2009
Received by editor(s) in revised form: October 6, 2009
Published electronically: February 2, 2010
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.