Abstract. The Lojasiewicz 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 |grad h(x, y)| ≥ c|(x, y)| λ holds near 0 ∈ C 2 for a certain c > 0. In the paper, we give an estimate of the Lojasiewicz exponent of grad h using information from the Newton diagram of h.We obtain the exact value of the exponent for non-degenerate series.1. Introduction. The main goal of this paper is to compute the Lojasiewicz exponent of the plane curve singularity using the Newton Polygon. Our theorem is the counterpart of the Kouchnirenko theorem [Kou] on the Milnor number in the case of two variables. We were inspired by the articles of Lichtin [Li] and Fukui [Fu] who proved some estimates for the Lojasiewicz exponent. However, our considerations are based on other ideas and enable us to calculate the Lojasiewicz exponent for non-degenerate power series. The organization of the paper is as follows. In Section 2 we present the main theorem with illustrative examples. Then we collect in Section 3 basic notions connected with the Newton diagrams. Our main reference is [BK]. A theorem concerning the Lojasiewicz exponent of a holomorphic mapping, determined by the pair of convergent power series, is given in Section 4. The theorem is helpful in a proof of the main result given in Section 6. The proof is preceded by the theorems which describe connections between the Newton diagram of the series h and the diagrams of its derivatives

Using the Newton algorithm we show how to compute all the polar quotients and their multiplicities of a plane curve f ¼ 0, where f is a formal power series of two variables over an algebraically closed field k with characteristic zero. The curve is not necessarily reduced.

Let $(l,f):(C^2,0)\rightarrow (C^2,0)$ be the germ of a holomorphic mapping such that $l=0$ is a smooth curve which is not a branch of the singular curve $f=0$. The direct image of the critical locus of this mapping is called the discriminant curve. In this paper we study the pairs $(l,f)$ for which the discriminant curve is non-degenerate in the Kouchnirenko sense

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