Strong minimality and the j-function

Freitag, James; Scanlon, Thomas

doi:10.48550/arxiv.1402.4588

Cited by 6 publications

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“…In the papers [8,6] the effective estimate of Theorem 2 has been used to derive estimates for some counting problems of a diophantine nature. In this section we illustrate our result by improving the estimates presented in these papers.…”

Section: Diophantine Applicationsmentioning

confidence: 99%

“…In [6] the authors, following a question of Mazur, consider the following problem: given an automorphism α of CP 1 and τ ∈ C, estimate the number of elliptic curves E ρ satisfying E ρ ∼ E τ and E α•ρ ∼ E α•τ . In particular they obtain the following estimate.…”

Section: Isogeny Classes Of Elliptic Curvesmentioning

confidence: 99%

“…However, in [6, Section 6.1] it is proven that this is indeed the case. Since Z is a set given by differential algebraic conditions, it is thus possible to estimate its size using the methods of [8], and this is carried out in [6] to give Theorem 33. We now apply our results, specifically Theorem 5, to estimate the size of Z.…”

Section: Isogeny Classes Of Elliptic Curvesmentioning

confidence: 99%

“…We refer the reader to [6] for the definition of a weakly special variety. As for the degree estimate, the proof of this corollary proceeds in a manner analogous to the proof of Theorem 33.…”

Section: Isogeny Classes Of Elliptic Curvesmentioning

confidence: 99%

“…ii. More recently, Theorem 2 has been used in [6] to derive bounds on the number of intersections between isogeny classes of elliptic curves and algebraic varieties. In particular, in the two-dimensional case this allowed the authors to produce doubly-exponential bounds for a counting problem due to Mazur.…”

Section: Introductionmentioning

confidence: 99%

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Bezout-type theorems for differential fields

Binyamini

2017

Compositio Math.

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Abstract. We prove analogs of the Bezout and the Bernstein-KushnirenkoKhovanskii theorems for systems of algebraic differential conditions over differentially closed fields. Namely, given a system of algebraic conditions on the first l derivatives of an n-tuple of functions, which admits finitely many solutions, we show that the number of solutions is bounded by an appropriate constant (depending singly-exponentially on n and l) times the volume of the Newton polytope of the set of conditions. This improves a doubly-exponential estimate due to Hrushovski and Pillay.We illustrate the application of our estimates in two diophantine contexts: to counting transcendental lattice points on algebraic subvarieties of semiabelian varieties, following Hrushovski and Pillay; and to counting the number of intersections between isogeny classes of elliptic curves and algebraic varieties, following Freitag and Scanlon. In both cases we obtain bounds which are singly-exponential (improving the known doubly-exponential bounds) and which exhibit the natural asymptotic growth with respect to the degrees of the equations involved.

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Section: Diophantine Applicationsmentioning

confidence: 99%

Section: Isogeny Classes Of Elliptic Curvesmentioning

confidence: 99%

Section: Isogeny Classes Of Elliptic Curvesmentioning

confidence: 99%

Section: Isogeny Classes Of Elliptic Curvesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bezout-type theorems for differential fields

Binyamini

2017

Compositio Math.

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Algebraic differential equations from covering maps

Scanlon

2018

Advances in Mathematics

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Let Y be a complex algebraic variety, G Y an action of an algebraic group on Y , U ⊆ Y (C) a complex submanifold, Γ < G(C) a discrete, Zariski dense subgroup of G(C) which preserves U , and π : U → X(C) an analytic covering map of the complex algebraic variety X expressing X(C) as Γ\U . We note that the theory of elimination of imaginaries in differentially closed fields produces a generalized Schwarzian derivative χ : Y → Z (where Z is some algebraic variety) expressing the quotient of Y by the action of the constant points of G. Under the additional hypothesis that the restriction of π to some set containing a fundamental domain is definable in an o-minimal expansion of the real field, we show as a consequence of the Peterzil-Starchenko o-minimal GAGA theorem that the prima facie differentially analytic relation χ := χ • π −1 is a well-defined, differential constructible function. The function χ nearly inverts π in the sense that for any differential field K of meromorphic functions, if a, b ∈ X(K) then χ(a) = χ(b) if and only if after suitable restriction there is some γ ∈ G(C) with π(γ · π −1 (a)) = b.

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Proof mining and effective bounds in differential polynomial rings

Simmons

Towsner

2019

Advances in Mathematics

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Using the functional interpretation from proof theory, we analyze nonconstructive proofs of several central theorems about polynomial and differential polynomial rings. We extract effective bounds, some of which are new to the literature, from the resulting proofs. In the process we discuss the constructive content of Noetherian rings and the Nullstellensatz in both the classical and differential settings. Sufficient background is given to understand the proof-theoretic and differentialalgebraic framework of the main results.1 A nonstandard approach. Invent. Math., 76(1): 77-91, 1984.

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Strong minimality and the j-function

Cited by 6 publications

References 0 publications

Bezout-type theorems for differential fields

Bezout-type theorems for differential fields

Algebraic differential equations from covering maps

Proof mining and effective bounds in differential polynomial rings

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