Computational Methods and Function Theory

, Volume 19, Issue 4, pp 687–716 | Cite as

On the Multipliers at Fixed Points of Quadratic Self-Maps of the Projective Plane with an Invariant Line

  • Adolfo GuillotEmail author
  • Valente Ramírez


This paper deals with holomorphic self-maps of the complex projective plane and the algebraic relations among the eigenvalues of the derivatives at the fixed points. These eigenvalues are constrained by certain index theorems such as the holomorphic Lefschetz fixed-point theorem. A simple dimensional argument suggests there must exist even more algebraic relations that the ones currently known. In this work we analyze the case of quadratic self-maps having an invariant line and obtain all such relations. We also prove that a generic quadratic self-map with an invariant line is completely determined, up to linear equivalence, by the collection of these eigenvalues. Under the natural correspondence between quadratic rational maps of \({\mathbb {P}}^2\) and quadratic homogeneous vector fields on \({\mathbb {C}}^3\), the algebraic relations among multipliers translate to algebraic relations among the Kowalevski exponents of a vector field. As an application of our results, we describe the sets of integers that appear as the Kowalevski exponents of a class of quadratic homogeneous vector fields on \({\mathbb {C}}^3\) having exclusively single-valued solutions.


Fixed point Self-map Projective space Multiplier Kowalevski exponent Painlevé test 

Mathematics Subject Classification

Primary 37C25 58C30 34M35 Secondary 37F10 14Q99 14N99 



The authors thank Jawad Snoussi and Javier Elizondo for helpful conversations, and Marco Abate and Masayo Fujimura for providing useful references.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Instituto de MatematicasUniversidad Nacional Autónoma de México, Ciudad UniversitariaMexico CityMexico
  2. 2.Institut de Recherche Mathématique de RennesUniversité de Rennes 1, UMR 6625RennesFrance

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