Note on the Rational Points of a Pfaff Curve

Pila, Jonathan

doi:10.1017/s0013091504000847

Cited by 14 publications

(38 citation statements)

References 10 publications

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“…The first choice improves the dependence on the degree d to polynomial in Heath-Brown's result [25] for hypersurfaces and Broberg's result [12, Theorem 1] (see also Marmon [31]). The second choice replaces an H ε factor by a power of log H, similar to the various results established for curves by Pila [42] and Salberger [47]. We remark that in the case of curves, Walsh [51] recently proved a result eliminating the log H factor altogether and it would be interesting to study whether this can be generalized to arbitrary dimension.…”

Section: 41mentioning

confidence: 69%

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Complex cellular structures

Binyamini

Novikov

2019

Ann. of Math. (2)

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We introduce the notion of a complex cell, a complexification of the cells/cylinders used in real tame geometry. For δ ∈ (0, 1) and a complex cell C we define its holomorphic extension C ⊂ C δ , which is again a complex cell. The hyperbolic geometry of C within C δ provides the class of complex cells with a rich geometric function theory absent in the real case. We use this to prove a complex analog of the cellular decomposition theorem of real tame geometry. In the algebraic case we show that the complexity of such decompositions depends polynomially on the degrees of the equations involved.Using this theory, we refine the Yomdin-Gromov algebraic lemma on C rsmooth parametrizations of semialgebraic sets: we show that the number of C r charts can be taken to be polynomial in the smoothness order r and in the complexity of the set. The algebraic lemma was initially invented in the work of Yomdin and Gromov to produce estimates for the topological entropy of C ∞ maps. For analytic maps our refined version, combined with work of Burguet, Liao and Yang, establishes an optimal refinement of these estimates in the form of tight bounds on the tail entropy and volume growth. This settles a conjecture of Yomdin who proved the same result in dimension two in 1991. A self-contained proof of these estimates using the refined algebraic lemma is given in an appendix by Yomdin.The algebraic lemma has more recently been used in the study of rational points on algebraic and transcendental varieties. We use the theory of complex cells in these two directions. In the algebraic context we refine a result of Heath-Brown on interpolating rational points in algebraic varieties. In the transcendental context we prove an interpolation result for (unrestricted) logarithmic images of subanalytic sets.

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Section: 41mentioning

confidence: 69%

“…It is natural to inquire whether one can in fact replace the C r -charts in the algebraic lemma with C ∞ charts with appropriate control over the derivatives. In [41] Pila introduced a notion of this type called mild parametrization and investigated its diophantine applications related to the Wilkie conjecture.…”

Section: Smooth Parametrizationsmentioning

confidence: 99%

Complex cellular structures

Binyamini

Novikov

2019

Ann. of Math. (2)

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“…Wilkie [14] studies integer points on curves in o-minimal structures; a result on the rational points of a pfaff curve is contained in [11].…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

Rational points on a subanalytic surface

Pila¹

2005

Annales De L’institut Fourier

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“…Primary 11G50; Secondary 03C64. This work was initiated while the authors were in residence at the Mathematical Sciences Research Institute (Berkeley) during the Model Theory, Arithmetic Geometry and Number Theory program (January 20, 2014 to May 23, 2014 o-minimal structures (see Binyamini and Novikov [5], Cluckers, Pila and Wilkie [11], Jones, Miller and Thomas [19], Jones and Thomas [20], Pila [24], [26], [27]). In this direction, and for more connections to logic, see Section 4. In this paper, we do not restrict attention to non-oscillatory sets (that is, sets that generate an o-minimal structure).…”

Section: Introductionmentioning

confidence: 99%

“…It is an easy exercise that ρ(sin(πx)) = +∞.) For X as in Propositions 1.1, 1.3 and 1.4, it follows from work of Pila [24] that every compact subset of X has finite order, but via proof that does not yield finite order for X itself. We remedy this by establishing the existence of compact connected K X ⊆ X such that ρ(X \K X ) < ∞, thus yielding ρ(X) = max(ρ(K X ), ρ(X \K X )) < +∞.…”

Section: Introductionmentioning

confidence: 99%

Points of bounded height on oscillatory sets

Comte

Miller

2017

The Quarterly Journal of Mathematics

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We show that transcendental curves in R n (not necessarily compact) have few rational points of bounded height provided that the curves are well behaved with respect to algebraic sets in a certain sense and can be parametrized by functions belonging to a specified algebra of infinitely differentiable functions. Examples of such curves include logarithmic spirals and solutions to Euler equations x 2 y ′′ +xy ′ + cy = 0 with c > 0.

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Note on the Rational Points of a Pfaff Curve

Abstract: Let X ⊂ R 2 be the graph of a Pfaffian function f in the sense of Khovanskii. Suppose that X is non-algebraic. This note gives an estimate for the number of rational points on X of height less than or equal to H; the estimate is uniform in the order and degree of f .

Cited by 14 publications

References 10 publications

Complex cellular structures

Complex cellular structures

Rational points on a subanalytic surface

Points of bounded height on oscillatory sets

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