Mildness and the density of rational points on certain transcendental curves. (English) Zbl 1220.03034

Jones, G. O.; Miller, D. J.; Thomas, M. E. M.

Notre Dame J. Formal Logic 52, No. 1, 67-74 (2011).

This paper concerns the density of rational points on certain real-analytic non-algebraic plane curves. The curves in question are restricted Pfaffian, namely, one-dimensional subsets of the plane definable in the o-minimal structure \({\mathbf R}_{\text{Res\,Pfaff}}\). The main theorem asserts that, for a non-algebraic curve \(X\) of this type, the number \(N(X,T)\) of rational points on the curve having both coordinates of height bounded by \(H\) is bounded by \(c(\log H)^\gamma\) for suitable constants \(c,\gamma\) depending on \(X\). The proof depends on the possibility of parameterising the curve in a suitable way, and is obtained by modifying the approach of J. Pila [Comment. Math. Univ. St. Pauli 55, No. 1, 1–8 (2006; Zbl 1129.11029)], where similar results are obtained for a more restricted type of curve definable in \({\mathbf R}_{\text{Pfaff}}\).

Reviewer: Jonathan Pila (Oxford)

MSC:

03C64	Model theory of ordered structures; o-minimality
11G99	Arithmetic algebraic geometry (Diophantine geometry)
11U09	Model theory (number-theoretic aspects)

Keywords:

Pfaffian functions; parameterization; rational point; o-minimal structure

Citations:

Zbl 1129.11029