Algebraic invariants, integrability, and meromorphic solutions

Abstract: Consider an autonomous algebraic ordinary differential equation of order higher than one. The aim of the talk is to address the following questions.

1. Does there exist an autonomous algebraic first-order ordinary differential equation compatible with the original equation?

2. If yes, how to find all such equations?

Bivariate polynomials producing autonomous algebraic first-order ordinary differential equations compatible with the equation under consideration are called algebraic invariants. The main difficulty in deriving irreducible algebraic invariants lies in the fact that the degrees of related bivariate polynomials are not known in advance.

Algebraic invariants are important from theoretical and practical point of views. In the two-dimensional case algebraic invariants are key objects in establishing Darboux and Liouvillian integrability of the original ordinary differential equation. In addition, algebraic invariants can be used to perform the classification of W-meromorphic solutions of ordinary differential equations. We shall pay some attention to these applications.

complex variablesdynamical systems

Audience: researchers in the topic

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CAvid: Complex Analysis video seminar

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