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

Abstract

We prove that every polynomial P ( x , y ) P(x,y) of degree d d has at most 2 ( d + 2 ) 12 2(d+2)^{12} zeros on the curve y = e x + sin ⁡ ( x ) , x > 0 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.