Parametrizing the branches of a curve by Puiseux series

2001 ◽  
pp. 125-146
Author(s):  
Gerd Fischer
Keyword(s):  
Puiseux Series

scholarly journals On the algebraicity of Puiseux series

2017 ◽  
Vol 30 (3) ◽  
pp. 589-620 ◽  
Author(s):  
Michel Hickel ◽  
Mickaël Matusinski
Keyword(s):  
Puiseux Series

Higher Order Tangents to Analytic Varieties along Curves

2003 ◽  
Vol 55 (1) ◽  
pp. 64-90 ◽  
Author(s):  
Rüdiger W. Braun ◽  
Reinhold Meise ◽  
B. A. Taylor
Keyword(s):  
Differential Operators ◽  
Necessary Conditions ◽  
Forthcoming Paper ◽  
Puiseux Series ◽  
Open Set ◽  
Partial Differential Operators ◽  
Real Analytic Functions ◽  
Pure Dimension ◽  
Analytic Varieties ◽  
Real Analytic

AbstractLet V be an analytic variety in some open set in which contains the origin and which is purely k-dimensional. For a curve γ in , defined by a convergent Puiseux series and satisfying γ(0) = 0, and d ≥ 1, define Vt := t−d(V − (t)). Then the currents defined by Vt converge to a limit current Tγ,d[V] as t tends to zero. Tγ,d[V] is either zero or its support is an algebraic variety of pure dimension k in . Properties of such limit currents and examples are presented. These results will be applied in a forthcoming paper to derive necessary conditions for varieties satisfying the local Phragmén-Lindelöf condition that was used by Hörmander to characterize the constant coefficient partial differential operators which act surjectively on the space of all real analytic functions on .

scholarly journals Rationality of dynamical canonical height

2018 ◽  
Vol 39 (9) ◽  
pp. 2507-2540
Author(s):  
LAURA DE MARCO ◽  
DRAGOS GHIOCA
Keyword(s):  
Elliptic Curve ◽  
Function Field ◽  
Rational Maps ◽  
Puiseux Series ◽  
Quadratic Maps ◽  
Canonical Height ◽  
Complete Answer ◽  
Irrational Values ◽  
Invent Math

We present a dynamical proof of the well-known fact that the Néron–Tate canonical height (and its local counterpart) takes rational values at points of an elliptic curve over a function field $k=\mathbb{C}(X)$, where $X$ is a curve. More generally, we investigate the mechanism by which the local canonical height for a map $f:\mathbb{P}^{1}\rightarrow \mathbb{P}^{1}$ defined over a function field $k$ can take irrational values (at points in a local completion of $k$), providing examples in all degrees $\deg f\geq 2$. Building on Kiwi’s classification of non-archimedean Julia sets for quadratic maps [Puiseux series dynamics of quadratic rational maps. Israel J. Math.201 (2014), 631–700], we give a complete answer in degree 2 characterizing the existence of points with irrational local canonical heights. As an application we prove that if the heights $\widehat{h}_{f}(a),\widehat{h}_{g}(b)$ are rational and positive, for maps $f$ and $g$ of multiplicatively independent degrees and points $a,b\in \mathbb{P}^{1}(\bar{k})$, then the orbits $\{f^{n}(a)\}_{n\geq 0}$ and $\{g^{m}(b)\}_{m\geq 0}$ intersect in at most finitely many points, complementing the results of Ghioca et al [Intersections of polynomials orbits, and a dynamical Mordell–Lang conjecture. Invent. Math.171 (2) (2008), 463–483].

scholarly journals Intrinsic Evolution of Truncated Puiseux Series on a Mixed-Signal Field-Programmable SoC

IEEE Access ◽  
2016 ◽  
Vol 4 ◽  
pp. 2863-2872 ◽  
Author(s):  
Vignesh Thangavel ◽  
Zi-Xia Song ◽  
Ronald F. Demara
Keyword(s):  
Puiseux Series ◽  
Mixed Signal ◽  
Field Programmable ◽  
Signal Field

scholarly journals Good reduction of Puiseux series and applications

2012 ◽  
Vol 47 (1) ◽  
pp. 32-63 ◽  
Author(s):  
Adrien Poteaux ◽  
Marc Rybowicz
Keyword(s):  
Good Reduction ◽  
Puiseux Series
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