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## The Converse of Abel's Theorem

In my thesis I investigate an algebraization problem. The simplest, but already nontrivial, problem in this direction is to find necessary and sufficient conditions for three graphs of smooth functions on a given interval to belong to an algebraic curve of degree three.

The analogous problems were raised by Lie and Darboux in connection with the

classification of surfaces of double translation; by Poincare and Mumford in connection with the Schottky problem; by Griffiths and Henkin in connection with a converse of Abel’s theorem; by Bol and Akivis in the connection with the algebraization problem in the theory of webs. Interestingly, the complex-analytic technique developed by Griftiths and Henkin for the holomorphic case failed to work in the real smooth setting.

In the thesis I develop a technique of, what I call, complex moments. Together with a

simple differentiation rule it provides a unified approach to all the algebraization problems considered so far (both complex-analytic and real smooth). As a result I prove two variants (’polynomial’ and ’rational’) of a converse of Abel’s theorem which significantly generalize results of Griffiths and Henkin. Already the ’polynomial’ case is nontrivial

leading to a new relation between the algebraization problem in the theory of webs and the converse of Abel’s theorem.

But, perhaps, the most interesting is the rational case as a new phenomenon occurs:

there are forms with logarithmic singularities on special algebraic varieties that satisfy the converse of Abel’s theorem. In the thesis I give a complete description of such varieties and forms.

Identifer | oai:union.ndltd.org:TORONTO/oai:tspace.library.utoronto.ca:1807/17785 |

Date | 24 September 2009 |

Creators | Kissounko, Veniamine |

Contributors | Khovanskii, Askold |

Source Sets | University of Toronto |

Language | en_ca |

Detected Language | English |

Type | Thesis |

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