Osaka Journal of Mathematics

Some generalizations of Halphen's equations

Adolfo Guillot

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Abstract

Halphen's equations are given by a remarkable polynomial vector field in $\mathbf{C}^{3}$ having only single-valued solutions, defined in domains bounded by a circle or by a line. By generalizing the Lie-theoretic principle behind Halphen's equations and borrowing some facts from the theory of deformations of Fuchsian groups, we exhibit a family of polynomial vector fields in $\mathbf{C}^{3}$ having only single-valued solutions. The solutions of vector fields within this family are defined in domains which had not been previously observed as domains of definition of solutions of polynomial vector fields in $\mathbf{C}^{3}$. For example, we obtain polynomial vector fields having solutions defined in domains that are bounded by a fractal curve.

Article information

Source
Osaka J. Math., Volume 48, Number 4 (2011), 1085-1094.

Dates
First available in Project Euclid: 11 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1326291219

Mathematical Reviews number (MathSciNet)
MR2871295

Zentralblatt MATH identifier
1251.32019

Subjects
Primary: 34M05: Entire and meromorphic solutions 34M15: Algebraic aspects (differential-algebraic, hypertranscendence, group- theoretical) 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]
Secondary: 17B66: Lie algebras of vector fields and related (super) algebras 57S30: Discontinuous groups of transformations

Citation

Guillot, Adolfo. Some generalizations of Halphen's equations. Osaka J. Math. 48 (2011), no. 4, 1085--1094. https://projecteuclid.org/euclid.ojm/1326291219


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