Skip to content
DraconiticResearch
Search Sources API
197,768 studies31 sourcesUpdated 21 min ago
PreprintarXiv licencearXiv

Generalized upper principal part of real planar polynomial vector fields

  • Thaís Maria Dalbelo
  • Regilene Oliveira
  • Otavio Henrique Perez
Open at sourcePDFDOICite

Abstract

The goal of this paper is to generalize recent results about the topological classification of the dynamics of a real planar polynomial vector field near infinity. Given a polynomial vector field X, using Newton polyhedra one can define its generalized upper principal part XΓU. By dropping some monomials of XΓU, we define its minimal generalized upper principal part XGU. We prove that there exist an open and dense set Ũ1 in the set of polynomial vector fields with Newton degenerate upper principal part satisfying the following property: if X∈Ũ1, then X and XGU are topologically equivalent near infinity, provided that XGU satisfies some additional non-degeneracy assumptions. Our techniques rely on toric compactification and the Normal Form Theorem.

Related

  1. Integrable Generators of Polynomial Vector Fields for Complex Classical Lie Groups
    Yiyang Jiang, Xudong Chen
    arXivPreprintarXiv licenceFull textAlgebra & number theoryAnalysis & PDEsarXiv
  2. The Lie algebra of polynomial vector fields on the affine space with constant divergence is $1.5$-generated
    Ivan Beldiev, Gleb Pogudin
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. Higher Order Structure along Nilpotent Eigendirections in Planar Vector Fields
    Roberto Albarran-García, Martha Alvarez Ramírez, Marco Polo García-Rivera
    arXivPreprintarXiv licenceFull textAnalysis & PDEsDynamical systemsarXiv
  4. Some remarks on polynomial vector fields in RR2
    Supriyo Jana, Soumen Sarkar
    arXivPreprintarXiv licenceFull textDynamical systemsarXiv
  5. Lyapunov stability of polynomial vector fields is undecidable
    Milan Korda
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsDynamical systemsarXiv

Cite

BibTeXRIS
BibTeX
@misc{dalbelo2026generalized,
  title = {{Generalized upper principal part of real planar polynomial vector fields}},
  author = {Thaís Maria Dalbelo and Regilene Oliveira and Otavio Henrique Perez},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04413},
  eprint = {2609.04413},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04413},
}
RIS
TY  - UNPB
TI  - Generalized upper principal part of real planar polynomial vector fields
AU  - Thaís Maria Dalbelo
AU  - Regilene Oliveira
AU  - Otavio Henrique Perez
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.04413
UR  - https://arxiv.org/abs/2609.04413
AB  - The goal of this paper is to generalize recent results about the topological classification of the dynamics of a real planar polynomial vector field near infinity. Given a polynomial vector field X, using Newton polyhedra one can define its generalized upper principal part X_Γ^U. By dropping some monomials of X_Γ^U, we define its minimal generalized upper principal part X_G^U. We prove that there exist an open and dense set Ũ_1 in the set of polynomial vector fields with Newton degenerate upper principal part satisfying the following property: if X∈Ũ_1, then X and X_G^U are topologically equivalent near infinity, provided that X_G^U satisfies some additional non-degeneracy assumptions. Our techniques rely on toric compactification and the Normal Form Theorem.
ER  -