The degree at infinity of the gradient of a polynomial in two real variables

Volume 87 / 2005

Maciej Sękalski Annales Polonici Mathematici 87 (2005), 229-235 MSC: 14P15, 32B10. DOI: 10.4064/ap87-0-19


Let $f:\mathbb{R}^2\to\mathbb{R}$ be a polynomial mapping with a finite number of critical points. We express the degree at infinity of the gradient $\nabla f$ in terms of the real branches at infinity of the level curves $\{f(x,y)=\lambda\}$ for some $\lambda\in\mathbb{R}$. The formula obtained is a counterpart at infinity of the local formula due to Arnold.


  • Maciej SękalskiDepartment of Mathematics
    Technical University
    Al. 1000LPP 7
    25-314 Kielce, Poland

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