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The map F0 is called the quasihomogeneous part of the semi-quasihomogeneous map F. Example. • , d - vn ). 3, formulated there for gradient maps, are valid for arbitrary quasihomogeneous and semi-quasihomogeneous maps ([16], [203]). The importance of the class of quasihomogeneous maps for the study of quasihomogeneous functions is explained by the fact that in that class one has greater freedom to construct homotopies and change variables than in the gradient case. Definition. The Poincare polynomial of a semi-quasihomogeneous map F of degree d and type v (where d; = DJN and Vi = NJN, with N, Ni> Dj E Z) is the polynomial PF(t) = Llliti, where Ili is the number of monomials of quasidegree i/N in an arbitrary monomial basis of the local algebra of F.

In what follows we shall consider quasihomogeneous functions of degree one with positive rational exponents. Such functions are automatically polynomials. The hyperplane L will be referred to as the diagonal. L cuts out segments of lengths 1/vi on the coordinate axes. Definition. , Jl(f) < 00. § 3. Reduction to Normal Forms 37 A necessary condition for the existence of a nondegenerate quasihomogeneous function with a prescribed set of exponents is that for each axis in the space of exponents there exists an exponent k E L at distance ~ 1 from that axis.

Equal to {O}). In other words, f satisfies Condition B if SJ a condition only on fo. = O. Thus, Condition B is actually Theorem B1. Iff satisfies Condition B, then Theorem T"p holds for r = p ~ 1. Definition. , than vJ Note that 9- is a finite-dimensional Lie algebra. L Definition. The series f = fo + ft + ... , equal to {O}). Note that Condition C, too, is imposed only on fo. Theorem CT. If f satisfies Condition C, then I; = A rolf. Corollary. Let f satisfy Condition C and let e t , e2' ... be quasihomogeneous polynomials of all possible degrees N + p, P ~ 0, whose images under the natural maps SlIp -+ A;' form bases in the spaces A;' of the spectral sequence.

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