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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.

11G50 Heights
11D75 Diophantine inequalities