scholarly journals The $ABC$-Conjecture implies uniform bounds on dynamical Zsigmondy sets

2020 ◽  
Vol 373 (7) ◽  
pp. 4627-4647 ◽  
Author(s):  
Nicole R. Looper
Keyword(s):  
Uniform Bounds ◽  
Abc Conjecture

Canonical Heights and Preperiodic Points for Certain Weighted Homogeneous Families of Polynomials

2018 ◽  
Vol 2019 (15) ◽  
pp. 4859-4879
Author(s):  
Patrick Ingram
Keyword(s):  
Lower Bound ◽  
Number Field ◽  
Uniform Bounds ◽  
Abc Conjecture ◽  
Canonical Height ◽  
Homogeneous Form ◽  
The Family

Abstract A family f of polynomials over a number field K will be called weighted homogeneous if and only if ft(z) = F(ze, t) for some binary homogeneous form F(X, Y) and some integer e ≥ 2. For example, the family zd + t is weighted homogeneous. We prove a lower bound on the canonical height, of the form \begin{align*} \hat{h}_{f_{t}}(z)\geq \varepsilon \max\!\left\{h_{\mathsf{M}_{d}}(f_{t}), \log|\operatorname{Norm}\mathfrak{R}_{f_{t}}|\right\},\end{align*} for values z ∈ K which are not preperiodic for ft. Here ε depends only on the number field K, the family f, and the number of places at which ft has bad reduction. For suitably generic morphisms $\varphi :\mathbb {P}^{1}\to \mathbb {P}^{1}$, we also prove an absolute bound of this form for t in the image of φ over K (assuming the abc Conjecture), as well as uniform bounds on the number of preperiodic points (unconditionally).

scholarly journals Stability of the Non-abelian X-ray Transform in Dimension $$\ge 3$$

2021 ◽  
Author(s):  
Jan Bohr
Keyword(s):  
Stability Estimate ◽  
Scattering Data ◽  
Regularity Conditions ◽  
Uniform Bounds ◽  
Ray Transform ◽  
Shallow Layer ◽  
Matrix Potential ◽  
Novel Method ◽  
Statistical Consistency ◽  
Appl Math

AbstractNon-abelian X-ray tomography seeks to recover a matrix potential $$\Phi :M\rightarrow {\mathbb {C}}^{m\times m}$$ Φ : M → C m × m in a domain M from measurements of its so-called scattering data $$C_\Phi $$ C Φ at $$\partial M$$ ∂ M . For $$\dim M\ge 3$$ dim M ≥ 3 (and under appropriate convexity and regularity conditions), injectivity of the forward map $$\Phi \mapsto C_\Phi $$ Φ ↦ C Φ was established in (Paternain et al. in Am J Math 141(6):1707–1750, 2019). The present article extends this result by proving a Hölder-type stability estimate. As an application, a statistical consistency result for $$\dim M =2$$ dim M = 2 (Monard et al. in Commun Pure Appl Math, 2019) is generalised to higher dimensions. The injectivity proof in (Paternain et al. in Am J Math 141(6):1707–1750, 2019) relies on a novel method by Uhlmann and Vasy (Invent Math 205(1):83–120, 2016), which first establishes injectivity in a shallow layer below $$\partial M$$ ∂ M and then globalises this by a layer stripping argument. The main technical contribution of this paper is a more quantitative version of these arguments, in particular, proving uniform bounds on layer depth and stability constants.

scholarly journals Uniform bounds on the number of rational points of a family of curves of genus 2

2004 ◽  
Vol 108 (2) ◽  
pp. 241-267 ◽  
Author(s):  
L. Kulesz ◽  
G. Matera ◽  
E. Schost
Keyword(s):  
Rational Points ◽  
Uniform Bounds ◽  
Family Of Curves ◽  
Genus 2

scholarly journals Strongly uniform bounds from semi-constructive proofs

2006 ◽  
Vol 141 (1-2) ◽  
pp. 89-107 ◽  
Author(s):  
Philipp Gerhardy ◽  
Ulrich Kohlenbach
Keyword(s):  
Uniform Bounds ◽  
Constructive Proofs
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