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Mild parameterization and the rational points of a Pfaff curve. (English) Zbl 1129.11029
Comment. Math. Univ. St. Pauli 55, No. 1, 1-8 (2006); erratum 55, No. 2, 231 (2006).
Let $$X\subset[-1,1]^2$$ be a Pfaff curve, so that $$X$$ is defined by solutions of a suitable system of differential equations. If $$X$$ is not semialgebraic it has been shown by the author and Wilkie [Duke Math. J. 133, No. 3, 591–616 (2006; Zbl 1217.11066)] that $$X$$ has $$O(H^\varepsilon)$$ rational points of height at most $$H$$; and it is conjectured that there are $$O_X((\log H)^c)$$ such points, for some constant $$c= c(X)$$.
The present paper proves this conjecture when $$X$$ admits “mild parameterization”, that is to say, the curve can be piecewise parameterized via functions satisfying $\sup_x\,|f^{(k)}(x)|\leq (Bk^c)^k\qquad\forall k\in\mathbb{N},$ with $$B$$, $$C$$ depending on $$f$$. The idea behind the proof can be traced back to the author’s joint work with E. Bombieri [Duke Math. J. 59, No. 2, 337–357 (1989; Zbl 0718.11048)].
An erratum notice retracts the claim made in remark 4.3.1.

##### MSC:
 11G50 Heights 11D75 Diophantine inequalities
##### Keywords:
Pfaff curve; rational points; bounded height; counting function