Painlev\'e's determinateness theorem extended to proper coverings

TL;DR

This paper extends Painlevé's determinateness theorem to complex first-order ODEs with multivalued solutions, using proper coverings to handle multivaluedness in the dependent variable.

Contribution

It introduces a novel extension of Painlevé's theorem to equations with multivalued known terms managed via proper coverings.

Findings

Extended Painlevé's theorem to multivalued cases

Established proper coverings as a tool for multivaluedness

Provided theoretical framework for complex ODEs with multivalued solutions

Abstract

We extend Painlev\'e's determinateness theorem to the case of first order ordinary differential equations in the complex domain with known terms allowed be multivalued in the dependent variable as well; multivaluedness is supposed to be resolved by proper coverings.