On the Łojasiewicz exponent of the gradient of a polynomial function
Volume 71 / 1999
Annales Polonici Mathematici 71 (1999), 211-239
DOI: 10.4064/ap-71-3-211-239
Abstract
Let $h = ∑ h_{αβ} X^αY^β$ be a polynomial with complex coefficients. The Łojasiewicz exponent of the gradient of h at infinity is the least upper bound of the set of all real λ such that $|grad h(x,y)| ≥ c|(x,y)|^λ$ in a neighbourhood of infinity in ℂ², for c > 0. We estimate this quantity in terms of the Newton diagram of h. Equality is obtained in the nondegenerate case.
