Let f:ℝ² → ℝ be a polynomial mapping with a finite number of critical points. We express the degree at infinity of the gradient ∇f in terms of the real branches at infinity of the level curves {f(x,y) = λ} for some λ ∈ ℝ. The formula obtained is a counterpart at infinity of the local formula due to Arnold.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap87-0-19,
author = {Maciej S\k ekalski},
title = {The degree at infinity of the gradient of a polynomial in two real variables},
journal = {Annales Polonici Mathematici},
volume = {85},
year = {2005},
pages = {229-235},
zbl = {1091.14017},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap87-0-19}
}
Maciej Sękalski. The degree at infinity of the gradient of a polynomial in two real variables. Annales Polonici Mathematici, Tome 85 (2005) pp. 229-235. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap87-0-19/