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Introduction

  • Joan C. Artés
  • Jaume Llibre
  • Alex C. Rezende
Chapter
  • 219 Downloads

Abstract

A vector field \(X: {\mathbb R}^2 \to {\mathbb R}^2\) of the form X = (P, Q) where \(P= \sum a_{ij}x^iy^j\) and \(Q= \sum b_{ij}x^iy^j\), 0 ≤ i + j ≤ n, is called a planar polynomial vector field of degree ≤ n. If ∑i+j=n(|aij| + |bij|) ≠ 0 then we say that X has degree n. In particular, polynomial vector fields of degree two are called quadratic vector fields. The M = (n + 1)(n + 2) real numbers aij, bij are called the coefficients of X. The space of these vector fields, endowed with the structure of an affine \({\mathbb R}^M\)-space in which X is identified with the M-tuple (a00, a10, …, a0n, b00, b10, …, b0n) of its coefficients, is denoted by \({\textsl {P}}_n ({\mathbb R}^2)\).

Notes

Acknowledgements

The first two authors are partially supported by MINECO grants number MTM2013-40998-P, an AGAUR grant number 2014SGR-568 and grants FP7-PEOPLE-2012-IRSES 318999 and 316338. The third author is partially supported by grant number CSF-PVE 88887.068602/2014-00 from CAPES-Brazil and a CNPq-Brazil grant number PDE 232336/ 2014-8.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Joan C. Artés
    • 1
  • Jaume Llibre
    • 1
  • Alex C. Rezende
    • 1
  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaCerdanyolaSpain

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