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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. (English) Zbl 0916.03026
It is proved that every polynomial $$P(x,y)$$ of degree $$d$$ has at most $$2(d+ 2)^{12}$$ zeros on the curve $$y= e^x+ \sin(x)$$, $$x>0$$.
As a consequence the authors obtain 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.

##### MSC:
 03C50 Models with special properties (saturated, rigid, etc.) 14P15 Real-analytic and semi-analytic sets 26E05 Real-analytic functions
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