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Let $h=\sum h_{\alpha \beta }{X}^{\alpha }{Y}^{\beta }$ 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 ${\left|gradh(x,y)\right|\ge c\left|(x,y)\right|}^{\lambda }$ 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.

The Łojasiewicz exponent of the gradient of a convergent power series h(X,Y) with complex coefficients is the greatest lower bound of the set of λ > 0 such that the inequality ${\left|gradh(x,y)\right|\ge c\left|(x,y)\right|}^{\lambda }$ holds near $0\in C^2$ for a certain c > 0. In the paper, we give an estimate of the Łojasiewicz exponent of grad h using information from the Newton diagram of h. We obtain the exact value of the exponent for non-degenerate series.