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

 

On the number of solutions of an algebraic equation on the curve $y = e^{x} +\sin x,\, x>0$,
and a consequence for o-minimal structures


Authors: Janusz Gwozdziewicz, Krzysztof Kurdyka and Adam Parusinski
Journal: Proc. Amer. Math. Soc. 127 (1999), 1057-1064
MSC (1991): Primary 32B20, 32C05, 14P15; Secondary 26E05, 03C99
MathSciNet review: 1476134
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that every polynomial $P(x,y)$ of degree $d$ has at most $2(d+2)^{12}$ zeros on the curve $y=e^{x}+\sin (x),\quad x>0 $. As a consequence we deduce that the existence of a uniform bound for the number of zeros of polynomials of a fixed degree on an analytic curve does not imply that this curve belongs to an o-minimal structure.


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Additional Information

Janusz Gwozdziewicz
Affiliation: Department of Mathematics, Technical University, Al. 1000LPP7, 25–314 Kielce, Poland
Email: matjg@eden.tu.kielce.pl

Krzysztof Kurdyka
Affiliation: Laboratoire de Mathématiques, Université de Savoie, Campus Scientifique 73 376 Le Bourget–du–Lac Cedex, France and Instytut Matematyki, Uniwersytet Jagielloński, ul. Reymonta 4 30–059 Kraków, Poland
Email: Krzysztof.Kurdyka@univ-savoie.fr

Adam Parusinski
Affiliation: Département de Mathématiques, Université d’Angers, 2, bd Lavoisier, 49045 Angers cedex 01, France
Email: parus@tonton.univ-angers.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04672-9
PII: S 0002-9939(99)04672-9
Keywords: Fewnomial, Khovansky theory, o-minimal structure
Received by editor(s): July 15, 1997
Communicated by: Steven R. Bell
Article copyright: © Copyright 1999 American Mathematical Society