Let h1(u),…, hk(u) be Laurent polynomials of m-variables and letbe a non-degenerate complete intersection variety. Such an intersection variety appears as an exceptional divisor of a resolution of non-degenerate complete intersection varieties with an isolated singularity at the origin (Ok4]).

Mixed functions are analytic functions in variables z1, . . . , zn and their conjugatesz1, . . . ,zn. We introduce the notion of Newton nondegeneracy for mixed functions and develop a basic tool for the study of mixed hypersurface singularities. We show the existence of a canonical resolution of the singularity, and the existence of the Milnor fibration under the strong non-degeneracy condition.

Polar weighted homogeneous polynomials are the class of special polynomials of real variable xi, yi, i = 1, . . . , n with zi = xi + √ −1yi which enjoys a " polar action". In many aspects, their behavior looks like that of complex weighted homogeneous polynomials. We study basic properties of hypersurfaces which are defined by polar weighted homogeneous polynomials.2000 Mathematics Subject Classification. 14J17, 32S25.

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