Quadratic differentials $(A(z-a)(z-b)/{(z-c)}^2)\mathrm{d}{z}^2$ and algebraic Cauchy transform

Atia, Mohamed Jalel, and Thabet, Faouzi. "Quadratic differentials $(A(z-a)(z-b)/(z-c)^{2}) {\rm d} z^{2}$ and algebraic Cauchy transform." Czechoslovak Mathematical Journal 66.2 (2016): 351-363. <http://eudml.org/doc/280094>.

@article{Atia2016,
abstract = {We discuss the representability almost everywhere (a.e.) in $\mathbb \{C\}$ of an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite number of isolated points. This brings us to the study of trajectories of the particular family of quadratic differentials $A(z-a)(z-b)\*(z-c)^\{-2\} \{\rm d\} z^\{2\}$. More precisely, we give a necessary and sufficient condition on the complex numbers $a$ and $b$ for these quadratic differentials to have finite critical trajectories. We also discuss all possible configurations of critical graphs.},
author = {Atia, Mohamed Jalel, Thabet, Faouzi},
journal = {Czechoslovak Mathematical Journal},
keywords = {algebraic equation; Cauchy transform; quadratic differential},
language = {eng},
number = {2},
pages = {351-363},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Quadratic differentials $(A(z-a)(z-b)/(z-c)^\{2\}) \{\rm d\} z^\{2\}$ and algebraic Cauchy transform},
url = {http://eudml.org/doc/280094},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Atia, Mohamed Jalel
AU - Thabet, Faouzi
TI - Quadratic differentials $(A(z-a)(z-b)/(z-c)^{2}) {\rm d} z^{2}$ and algebraic Cauchy transform
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 2
SP - 351
EP - 363
AB - We discuss the representability almost everywhere (a.e.) in $\mathbb {C}$ of an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite number of isolated points. This brings us to the study of trajectories of the particular family of quadratic differentials $A(z-a)(z-b)\*(z-c)^{-2} {\rm d} z^{2}$. More precisely, we give a necessary and sufficient condition on the complex numbers $a$ and $b$ for these quadratic differentials to have finite critical trajectories. We also discuss all possible configurations of critical graphs.
LA - eng
KW - algebraic equation; Cauchy transform; quadratic differential
UR - http://eudml.org/doc/280094
ER -