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

Gwoździewicz, Janusz; Kurdyka, Krzysztof; Parusiński, Adam

doi:10.1090/S0002-9939-99-04672-9

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
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by Janusz Gwoździewicz, Krzysztof Kurdyka and Adam Parusiński PDF

Proc. Amer. Math. Soc. 127 (1999), 1057-1064 Request permission

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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