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\(C^l-{\mathcal G}_v\)-determinacy of weighted homogeneous function germs on weighted homogeneous analytic varieties. (English) Zbl 1154.58021
The authors provide estimates on the degree of \( C^l-{\mathcal G}_V \)-determinacy (where \( \mathcal G \) is one of Mather’s groups \( \mathcal R \) or \( \mathcal K \)) of weighted homogeneous function germs defined on a weighted homogeneous analytic variety \( V \) and satisfying a suitable Łojasiewicz condition. They give an explicit order such that the \( C^l \) geometric structure of a weighted homogeneous polynomial is preserved under higher order perturbations. This extends results on \( C^l-{\mathcal K} \) determinacy of homogeneous functions germs given by M. A. Soares Ruas [Math. Scand. 59, No. 1, 59–70 (1986; Zbl 0632.58013)] and M. A. Soares Ruas and M. J. Saia [Hokkaido Math. J. 26, No. 1, 89–99 (1997; Zbl 0873.58010)].
MSC:
58K40 Classification; finite determinacy of map germs
58A35 Stratified sets
32C05 Real-analytic manifolds, real-analytic spaces
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