On the Łojasiewicz exponent of the gradient of a polynomial function

Volume 71 / 1999

Andrzej Lenarcik Annales Polonici Mathematici 71 (1999), 211-239 DOI: 10.4064/ap-71-3-211-239


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.


  • Andrzej Lenarcik

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