Authors: Dragović, Vladimir 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Multivalued hyperelliptic continued fractions of generalized Halphen type
Journal: International Mathematics Research Notices
Volume: 2009
Issue: 10
First page: 1891
Last page: 1932
Issue Date: 1-Dec-2009
Rank: M21
ISSN: 1073-7928
DOI: 10.1093/imrn/rnp005
We introduce and study higher genera generalizations of the Halphen theory of continued fractions. The basic notion we start with is hyperelliptic Halphen (HH) element √X 2g+2 - √Y 2/x - y, depending on parameter y, where X 2g+2 is a polynomial of degree 2g + 2 and Y 2g+2 = X 2g+2(y). We study regular and irregular HH elements, their continued fraction developments, and some basic properties of such developments such as even and odd symmetry and periodicity. There is a 2 ↔ g + 1 dynamics, which lies in the basis of the developed continued fractions theory. We give two geometric realizations of this dynamics. The first one deals with nets of polynomials and with polygons circumscribed about a conic K. The dynamics is realized as a path of polygons of g + 1 sides inscribed in a curve B of degree 2g and circumscribed about the conic K obtained by successive moves, so called flips along edges. The second geometric realization leads to a new interpretation of generalized Jacobians of hyperelliptic curves.
Publisher: Oxford University Press
Project: Serbian Ministry of Science and Technology, Project Geometry and Topology of Manifolds and Integrable Dynamical Systems

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