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

Czechoslovak Mathematical Journal

Faouzi Thabet¹  Mohamed Jalel Atia² 

[1]Institut Supérieur des Sciences Appliquées et Technologies de Gabs, University of Gabs, Avenue Omar Ibn El Khattab, 6029 Gabs, Tunisia

[2]University of Gabs, Faculté des Sciences de Gabs, Cité Erriadh Zrig 6072, Gabs, Tunisia

关键词: algebraic equation;    Cauchy transform;    quadratic differential;   

DOI  :  

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

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 $A(z-a)(z-b)\*(z-c)^{-2}  d z^2$. 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.

【 授权许可】

Unknown