On the Lojasiewicz exponent of the gradient of a polynomial function

Abstract

Let h = \sum h_{\alpha \beta} X^\alpha Y^\beta be a polynomial with complex coefficients. The Lojasiewicz exponent of the gradient of h at infinity is the upper bound of the set of all real \lambda such that |grad h(x, y)| >= c|(x,y)|^\lambda in a neighbourhood of infinity in C^2, for c > 0. We estimate this quantity in terms of the Newton diagram of h. The equality is obtained in the nondegenerate case.