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

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.