Note on the rational points of a Pfaff curve. (English) Zbl 1097.11037

Pila, Jonathan

Proc. Edinb. Math. Soc., II. Ser. 49, No. 2, 391-397 (2006).

Let \(f\) be a real analytic transcendental function in several variables, with graph \(X\). The author seeks bounds \(O(H^\varepsilon)\) for the number of rational points on \(X\), of height at most \(H\), for arbitrarily small \(\varepsilon\). He previously [Ann. Inst. Fourier 55, No. 5, 1501–1516 (2005; Zbl 1121.11032)] proved such a bound when \(f\) is compactly supported, the order constant depending on \(f\) and \(\varepsilon\). For the more difficult case of non-compact support he now handles Pfaffian functions, which, roughly speaking, are those definable by suitable systems of differential equations with polynomial coefficients. He shows that if \(H\geq H_0\) then there are at most \(\exp(5\sqrt{\log H})\) points of height up to \(H\). Here \(H_0\) depends only on the order and degree of \(f\) (suitably defined).

Reviewer: Roger Heath-Brown (Oxford)

MSC:

11J99	Diophantine approximation, transcendental number theory
11D99	Diophantine equations
11G50	Heights

Keywords:

analytic function; Pfaffian function; height