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Integrable systems blackboard seminar
Integrable systems blackboard seminar
Construction of Small Functions, Finiteness Properties, and Beyond
Construction of Small Functions, Finiteness Properties, and Beyond
Organizers
Speaker
Techheang Meng
Time
Monday, November 16, 2026 3:30 PM - 4:30 PM
Venue
A7-201
Abstract
Given an ODE, it is well known that its solutions can be expressed in terms of first integrals, provided that a sufficient number of first integrals exists. Examples include Liouville integrability of Hamiltonian systems and the existence of (n-1) first integrals for a generic (n)th-order ODE. When there are not enough first integrals, however, certain particular solutions, such as meromorphic solutions, may still be found.
Eremenko showed that if an ODE admits only finitely many Laurent series expansions (from the Painlev\'e test), then it admits only meromorphic solutions that are elliptic functions or their degenerations. Halburd and Wang extended this idea by allowing arbitrary parameters to appear in the Laurent series, provided that they occur only in sufficiently high-order terms.
In this talk, we will use this idea to construct a small function that plays a role similar to that of a first integral for each meromorphic solution of a given ODE. Using these small functions, we will characterize the meromorphic solutions and discuss further ideas beyond these methods.
Eremenko showed that if an ODE admits only finitely many Laurent series expansions (from the Painlev\'e test), then it admits only meromorphic solutions that are elliptic functions or their degenerations. Halburd and Wang extended this idea by allowing arbitrary parameters to appear in the Laurent series, provided that they occur only in sufficiently high-order terms.
In this talk, we will use this idea to construct a small function that plays a role similar to that of a first integral for each meromorphic solution of a given ODE. Using these small functions, we will characterize the meromorphic solutions and discuss further ideas beyond these methods.