By V. I. Arnold, V. V. Goryunov, O. V. Lyashko, V. A. Vasil’ev (auth.)

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Medelingen van Het Wiskundig Genootschap, 1995

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ecu Mathematical Society publication, 1994

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IMS Bulletin, 1995

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**Example text**

Jo). Moreover, as follows from the next theorem, the multiplicity J1. is the same for all semiquasihomogeneous functions with a fixed degree and a fixed type of quasihomogeneity. o of a semi-quasihomogeneous function J is equal Theorem (see [37]). The number oj monomials oj a given degree ~ in a basis oj the local algebra oj a semi-quasihomogeneous Junction J does not depend on the choice oj the monomial basis Jor the local algebra and is the same Jor all semi-quasihomogeneous Junctions oj degree d and oj a given type oj quasihomogeneity v.

C, 0) is exactly the local algebra of the gradient map x H VJ(x)). Definition. A map F is said to be nondegenerate if the C-dimension J1. = dimcQF of its local algebra QF is finite, in which case J1. is called the multiplicity oj F at O. §3. Reduction to Normal Forms 39 Definition. F is said to be a semi-quasihomogeneous map ifF = Fo + F', where Fo is a nondegenerate quasihomogeneous map and the order of each component of F' is higher that the order of the corresponding component of Fo. The map F0 is called the quasihomogeneous part of the semi-quasihomogeneous map F.

Definition. The group of d-quasijets of type v is the quotient group of the group of diffeomorphisms Go by the subgroup Gd + of diffeomorphisms of order higher than d: Jd = GO/Gd +· Remark. In the ordinary homogeneous case our numbering differs from the standard one by 1: for us Jo is the group of I-jets, and so on. Jd acts as a group of linear transformations on the space A/A d + of d-quasijets of functions. A special importance is attached to the group Jo, which is the quasihomogeneous generalization of the general linear group.