scholarly journals A note on rational points near planar curves

2017 ◽  
Vol 177 (4) ◽  
pp. 393-396
Author(s):  
Sam Chow
Keyword(s):  
Rational Points ◽  
Planar Curves

scholarly journals Counting rational points near planar curves

2014 ◽  
Vol 165 (1) ◽  
pp. 91-100 ◽  
Author(s):  
Ayla Gafni
Keyword(s):  
Rational Points ◽  
Planar Curves ◽  
Counting Rational Points

scholarly journals Diophantine approximation on planar curves and the distribution of rational points (with an Appendix by R. C. Vaughan)

2007 ◽  
Vol 166 (2) ◽  
pp. 367-426 ◽  
Author(s):  
Victor Beresnevich ◽  
Detta Dickinson ◽  
Sanju Velani ◽  
Robert Vaughan
Keyword(s):  
Diophantine Approximation ◽  
Rational Points ◽  
Planar Curves

scholarly journals Affine Subspaces of Curvature Functions from Closed Planar Curves

2021 ◽  
Vol 76 (2) ◽  
Author(s):  
Leonardo Alese
Keyword(s):  
Arc Length ◽  
Planar Curves ◽  
Affine Subspaces ◽  
Planar Curve ◽  
Real Functions ◽  
Discrete Counterpart ◽  
Equivalent Conditions

AbstractGiven a pair of real functions (k, f), we study the conditions they must satisfy for $$k+\lambda f$$ k + λ f to be the curvature in the arc-length of a closed planar curve for all real $$\lambda $$ λ . Several equivalent conditions are pointed out, certain periodic behaviours are shown as essential and a family of such pairs is explicitely constructed. The discrete counterpart of the problem is also studied.

scholarly journals Clothoid fitting and geometric Hermite subdivision

2021 ◽  
Vol 47 (4) ◽  
Author(s):  
Ulrich Reif ◽  
Andreas Weinmann
Keyword(s):  
Building Block ◽  
Numerical Results ◽  
Normal Vector ◽  
Subdivision Schemes ◽  
Planar Curves ◽  
Nonlinear Subdivision ◽  
New Strategy ◽  
Vector Information ◽  
Hermite Subdivision

AbstractWe consider geometric Hermite subdivision for planar curves, i.e., iteratively refining an input polygon with additional tangent or normal vector information sitting in the vertices. The building block for the (nonlinear) subdivision schemes we propose is based on clothoidal averaging, i.e., averaging w.r.t. locally interpolating clothoids, which are curves of linear curvature. To this end, we derive a new strategy to approximate Hermite interpolating clothoids. We employ the proposed approach to define the geometric Hermite analogues of the well-known Lane-Riesenfeld and four-point schemes. We present numerical results produced by the proposed schemes and discuss their features.

scholarly journals Counting and equidistribution in quaternionic Heisenberg groups

2021 ◽  
pp. 1-38
Author(s):  
JOUNI PARKKONEN ◽  
FRÉDÉRIC PAULIN
Keyword(s):  
Hyperbolic Geometry ◽  
Quaternion Algebra ◽  
Rational Points ◽  
Hyperbolic Spaces ◽  
Heisenberg Groups ◽  
Arithmetic Groups ◽  
Hermitian Forms ◽  
Dimension 2 ◽  
Counting Formula ◽  
The Relationship

Abstract We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

scholarly journals Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

2021 ◽  
Author(s):  
Tim Browning ◽  
Shuntaro Yamagishi
Keyword(s):  
Circle Method ◽  
Rational Points ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Waring’S Problem ◽  
Waring Problem ◽  
Waring's Problem ◽  
Main Conjecture ◽  
Higher Dimensional ◽  
Thin Set

AbstractWe study the density of rational points on a higher-dimensional orbifold $$(\mathbb {P}^{n-1},\Delta )$$ ( P n - 1 , Δ ) when $$\Delta $$ Δ is a $$\mathbb {Q}$$ Q -divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.

scholarly journals Le complémentaire des puissances -ièmes dans un corps de nombres est un ensemble diophantien

2015 ◽  
Vol 151 (10) ◽  
pp. 1965-1980 ◽  
Author(s):  
Jean-Louis Colliot-Thélène ◽  
Jan Van Geel
Keyword(s):  
Prime Number ◽  
Rational Points ◽  
Parameter Family ◽  
Symmetric Power ◽  
Symmetric Powers

For $n=2$ the statement in the title is a theorem of B. Poonen (2009). He uses a one-parameter family of varieties together with a theorem of Coray, Sansuc and one of the authors (1980), on the Brauer–Manin obstruction for rational points on these varieties. For $n=p$, $p$ any prime number, A. Várilly-Alvarado and B. Viray (2012) considered analogous families of varieties. Replacing this family by its $(2p+1)$th symmetric power, we prove the statement in the title using a theorem on the Brauer–Manin obstruction for rational points on such symmetric powers. The latter theorem is based on work of one of the authors with Swinnerton-Dyer (1994) and with Skorobogatov and Swinnerton-Dyer (1998), work generalising results of Salberger (1988).

Sign in / Sign up

Export Citation Format

Share Document