| Volume 1, Number 2, 2011, Pages 173-182 DOI：10.11948/2011011 |
| On the number of \(n\)-dimensional invariant spheres in polynomial vector fields of \(\mathbb{C}^{n+1}\) |
| Yudy Bola\(\tilde{n}\)os,Jaume Llibre |
| Keywords:polynomial vector fields, invariant spheres, invariant circles, extactic algebraic hypersurface |
| Abstract: |
| We study the polynomial vector fields \(\mathcal{X}= \displaystyle \sum_{i=1}^{n+1} P_i(x_1,\ldots,x_{n+1}) \frac{\partial}{\partial x_i}\) in \(\mathbb{C}^{n+1}\) with \(n\geq 1\) . Let \(m_i\) be the degree of the polynomial \(P_i\). We call \((m_1,\ldots,m_{n+1})\) the degree of \(\mathcal{X}\). For these polynomial vector fields \(\mathcal{X}\) and in function of their degree we provide upper bounds, first for the maximal number of invariant \(n\)-dimensional spheres, and second for the maximal number of \(n\)-dimensional concentric invariant spheres. |
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