Existence and nonexistence of Puiseux inverse integrating factors in analytic monodromic singularities

In this work, we present some criteria about the existence and nonexistence of both Puiseux inverse integrating factors V$V$ and Puiseux first integrals H$H$ for planar analytic vector fields having a monodromic singularity. These functions are a wide generalization of their formal R[[x,y]]$\mathbb...

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Bibliographic Details
Published in:Studies in applied mathematics (Cambridge) 2024-08, Vol.153 (2), p.n/a
Main Authors: García, Isaac A., Giné, Jaume, Rodero, Ana Livia
Format: Article
Language:English
Subjects:
Cartesian coordinates
center‐focus problem
Fields (mathematics)
first integral
inverse integrating factor
Mathematical analysis
Singularity (mathematics)
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Summary:In this work, we present some criteria about the existence and nonexistence of both Puiseux inverse integrating factors V$V$ and Puiseux first integrals H$H$ for planar analytic vector fields having a monodromic singularity. These functions are a wide generalization of their formal R[[x,y]]$\mathbb {R}[[x,y]]$ or algebraic counterpart in Cartesian coordinates (x,y)$(x,y)$. We prove that none of the functions H$H$ and V$V$ can be used to characterize degenerate centers although the existence of H$H$ is a sufficient center condition.
ISSN:0022-2526
1467-9590
DOI:10.1111/sapm.12724