On the Invariance of the Real Milnor Number under Asymptotically Lipschitz Equivalence
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908344317378560 |
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| author | de Omena, Raphael Sampaio, José Edson Souza, Emanoel |
| author_facet | de Omena, Raphael Sampaio, José Edson Souza, Emanoel |
| contents | We investigate sufficient conditions for the invariance of the real Milnor number under $\mathcal{R}$-bi-Lipschitz equivalence for function-germs $ f, g \colon (\mathbb{R}^n, 0) \to (\mathbb{R}, 0) $. More generally, we explore its invariance within the extended framework of $\mathcal{R}$-asymptotically Lipschitz equivalence. To this end, we introduce the $α$-derivative, which provides a natural setting for studying asymptotic growth. Additionally, we discuss the implications of our results in the context of $C^k$ and $C^{\infty}$ equivalences, establishing sufficient conditions for the real Milnor number to remain invariant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_00166 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Invariance of the Real Milnor Number under Asymptotically Lipschitz Equivalence de Omena, Raphael Sampaio, José Edson Souza, Emanoel Algebraic Geometry Metric Geometry We investigate sufficient conditions for the invariance of the real Milnor number under $\mathcal{R}$-bi-Lipschitz equivalence for function-germs $ f, g \colon (\mathbb{R}^n, 0) \to (\mathbb{R}, 0) $. More generally, we explore its invariance within the extended framework of $\mathcal{R}$-asymptotically Lipschitz equivalence. To this end, we introduce the $α$-derivative, which provides a natural setting for studying asymptotic growth. Additionally, we discuss the implications of our results in the context of $C^k$ and $C^{\infty}$ equivalences, establishing sufficient conditions for the real Milnor number to remain invariant. |
| title | On the Invariance of the Real Milnor Number under Asymptotically Lipschitz Equivalence |
| topic | Algebraic Geometry Metric Geometry |
| url | https://arxiv.org/abs/2505.00166 |