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We 0 0 0 0 define x˛ y ˇ z x˛ y ˇ z if x˛ > x˛ in the purely lexicographic order, or 0 0 else if z > z in the purely lexicographic order, or else if y ˇ > y ˇ in the degree lexicographic P order. 1) holds. Pt Note that G contains in particular those rewriting relations Ái Áj kD1 ´i qij k y1 ; : : : ; yn / which express the Hironaka decompositions of all quadratic monomials in the Á’s. C n /. Our algorithm will be set up so that it generates an explicit Hironaka decomposition for the invariant ring CŒx€ .

O. p. are called primary invariants, while the Áj are called secondary invariants. Áj /. Note that for a given group € there are many different Hironaka decompositions. Also the degrees of the primary and secondary invariants are not unique. C 1 /, then we have CŒx€ D CŒx D CŒx 2  ˚ x CŒx 2  D CŒx 3  ˚ x CŒx 3  ˚ x 2 CŒx 3  D : : : : But there is also a certain uniqueness property. Suppose that we already know the primary invariants or at least their degrees di , i D 1; : : : ; n. Then the number t of secondary invariants can be computed from the following explicit formula.

X1 B / C ranges over €. xn B / where press each power sum Se as a polynomial function in the first j€j power sums S1 ; S2 ; : : : ; Sj€j . Such a representation of Se shows that all u-coefficients are actually polynomial functions in the u-coefficients of S1 ; S2 ; : : : ; Sj€j . This argument proves that the invariants Je with jej > j€j are contained in the subring C fJe W jej Ä j€jg . We have noticed above that every invariant is a C-linear combination of the special invariants Je . This implies that CŒx€ D C fJe W jej Ä j€jg : The set of integer vectors e 2 N n with jej Ä j€j has cardinality nCj€j n .

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