Topological classification of generic real rational functions

Sergei Natanzon; Boris Shapiro; Alek Vainshtein

doi:10.1142/S021821650200213X

Topological classification of generic real rational functions

Sergei Natanzon, Boris Shapiro, Alek Vainshtein

Research output: Contribution to journal › Article › peer-review

Abstract

To any real rational function with generic ramification points we assign a combinatorial object, called a garden, which consists of a weighted labeled directed planar chord diagram and of a set of weighted rooted trees each corresponding to a face of the diagram. We prove that any garden corresponds to a generic real rational function, and that equivalent functions have equivalent gardens.

Original language	English
Pages (from-to)	1063-1075
Number of pages	13
Journal	Journal of Knot Theory and its Ramifications
Volume	11
Issue number	7
DOIs	https://doi.org/10.1142/S021821650200213X
State	Published - Nov 2002

Bibliographical note

Funding Information:
Partly supported by the INTAS grant 00-0259 and by the RFFI grant 01-01-00739.

Funding Information:
The authors are sincerely grateful to the Max-Planck Mathematical Institute in Bonn for the financial support and excellent research atmosphere during the fall of 2000 when this project was started. The second author wants to acknowledge the hospitality of IHES, Paris in January 2001 during the final stage of preparation of the manuscript.

Keywords

Generic ramification
Hurwitz numbers
Real rational functions
Topological invariants

ASJC Scopus subject areas

Algebra and Number Theory

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10.1142/S021821650200213X

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keywords = "Generic ramification, Hurwitz numbers, Real rational functions, Topological invariants",

author = "Sergei Natanzon and Boris Shapiro and Alek Vainshtein",

note = "Funding Information: Partly supported by the INTAS grant 00-0259 and by the RFFI grant 01-01-00739. Funding Information: The authors are sincerely grateful to the Max-Planck Mathematical Institute in Bonn for the financial support and excellent research atmosphere during the fall of 2000 when this project was started. The second author wants to acknowledge the hospitality of IHES, Paris in January 2001 during the final stage of preparation of the manuscript.",

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AU - Shapiro, Boris

AU - Vainshtein, Alek

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AB - To any real rational function with generic ramification points we assign a combinatorial object, called a garden, which consists of a weighted labeled directed planar chord diagram and of a set of weighted rooted trees each corresponding to a face of the diagram. We prove that any garden corresponds to a generic real rational function, and that equivalent functions have equivalent gardens.

KW - Generic ramification

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KW - Topological invariants

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Topological classification of generic real rational functions

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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