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

Gwozdziewicz, Janusz; Kurdyka, Krzysztof; Parusinski, Adam

Proc. Am. Math. Soc. 127, No. 4, 1057-1064 (1999).

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.

Reviewer: Viorel Vâjâitu (Bucureşti)

MSC:

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

Keywords:

algebraic equation; fewnomial; Khovansky theory; o-minimal structure; analytic curve