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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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