Newton Polygon Relative to an Arc

The notion of Newton polygon is well-known. We deene a generali-sation and apply it to study polar curves, Lojasiewicz exponents, singularities at innnity of complex polynomials (Ha Huy Vui's theorem), and-constant deformations. Philosophically speaking, the Newton polygon relative to an arc exposes f in a horn neighborhood of. The gradient of a function behaves erratically in the process of blowing up. Our method indicates how it can be handled without resort to blow-ups. Throughout this paper let f(x; y) denote a germ of holomorphic function with Taylor expansion: We shall assume f(x; y) is mini-regular in x of order k in the sense that H k (1; 0) 6 = 0. (This can be achieved by a linear transformation x 0 = x, y 0 = y + cx, c a generic constant.) By a fractional (convergent) power series we mean a series of the form < are positive integers, having no common divisor, such that (t N) has positve radius of convergence. We can identify with the analytic arc : x = c 1 t n 1 + c 2 t n 2 + ; y = t N , jtj small, which is not tangent to the x-axis (since n 1 =N 1). For each c ij 6 = 0, let us plot a dot at (i; j=N), called a Newton dot. The set of Newton dots is called the Newton diagram. They generate a convex hull, whose boundary is 1991 Mathematics Subject Classiication. 32S05, 32S15, 14H20. The paper was prepared during the rst author's stay in Angers. We would like to thank PRIA (P^ ole de Recherche et d'Innovation a Angers) for the nancial support which made this visit possible.