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Darboux theory of integrability for real polynomial vector fields on the $n-$dimensional ellipsoid

J. Llibre, Adrian C. Murza

Abstract

We extend to the $n$-dimensional ellipsoid contained in $\R^{n+1},$ the Darboux theory of integrability for polynomial vector fields in the $n$-dimensional sphere (Llibre et al., 2018). New results on the maximum number of invariant parallels and meridians of polynomial vector fields $\X$ on the invariant $n-$dimensional ellipsoid, as a function of its degree, are provided. Our results extend the known result on the upper bound for the number of invariant hyperplanes that a polynomial vector field $\Y$ in $\R^n$ can have in function of the degree of $\Y$.

Darboux theory of integrability for real polynomial vector fields on the $n-$dimensional ellipsoid

Abstract

We extend to the -dimensional ellipsoid contained in the Darboux theory of integrability for polynomial vector fields in the -dimensional sphere (Llibre et al., 2018). New results on the maximum number of invariant parallels and meridians of polynomial vector fields on the invariant dimensional ellipsoid, as a function of its degree, are provided. Our results extend the known result on the upper bound for the number of invariant hyperplanes that a polynomial vector field in can have in function of the degree of .

Paper Structure

This paper contains 3 sections, 8 theorems, 15 equations.

Table of Contents

  1. Introduction and statement of the main results
  2. Darboux theory of integrability in $\mathbb{R}^n$ and invariant hyperplanes
  3. Darboux theory of integrability in $\mathbb{E}^n$ and invariant parallels and meridians

Key Result

Theorem 1

Suppose that a polynomial vector field $\mathcal{X}$ defined in $\mathbb R^n$ of degree $m= (m_1,\ldots,m_n)$ admits $p$ invariant algebraic hypersurfaces $f_i=0$ with cofactors $K_i$ for $i=1,\ldots,p$, and $q$ exponential factors $F_j=\exp(g_j/h_j)$ with cofactors $L_j$ for $j=1,\ldots,q$. Then th

Theorems & Definitions (13)

  • Theorem 1
  • Proposition 2
  • Theorem 3
  • Remark 4
  • Theorem 5
  • Theorem 6
  • proof
  • Theorem 7
  • proof
  • Proposition 8
  • ...and 3 more