Geodesic
Quadratic differentials $(A(z-a)(z-b)/(z-c)^{2}) {\rm d} z^{2}$ and algebraic Cauchy transform
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 351-363 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.1007/s10587-016-0260-3
Classification : 28A99, 30L05
Keywords: algebraic equation; Cauchy transform; quadratic differential 
@article{10_1007_s10587_016_0260_3,
     author = {Atia, Mohamed Jalel and Thabet, Faouzi},
     title = {Quadratic differentials $(A(z-a)(z-b)/(z-c)^{2}) {\rm d} z^{2}$ and algebraic {Cauchy} transform},
     journal = {Czechoslovak Mathematical Journal},
     pages = {351--363},
     year = {2016},
     volume = {66},
     number = {2},
     doi = {10.1007/s10587-016-0260-3},
     mrnumber = {3519606},
     zbl = {06604471},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0260-3/}
}
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
SP  - 351
EP  - 363
VL  - 66
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0260-3/
DO  - 10.1007/s10587-016-0260-3
LA  - en
ID  - 10_1007_s10587_016_0260_3
ER  - 
%0 Journal Article
%A Atia, Mohamed Jalel
%A Thabet, Faouzi
%T Quadratic differentials $(A(z-a)(z-b)/(z-c)^{2}) {\rm d} z^{2}$ and algebraic Cauchy transform
%J Czechoslovak Mathematical Journal
%D 2016
%P 351-363
%V 66
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0260-3/
%R 10.1007/s10587-016-0260-3
%G en
%F 10_1007_s10587_016_0260_3
Atia, Mohamed Jalel; Thabet, Faouzi. Quadratic differentials $(A(z-a)(z-b)/(z-c)^{2}) {\rm d} z^{2}$ and algebraic Cauchy transform. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 351-363. doi: 10.1007/s10587-016-0260-3

Cité par Sources :