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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2503.06022 |
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| _version_ | 1866910865079402496 |
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| author | Alvarez, Sergio |
| author_facet | Alvarez, Sergio |
| contents | We investigate the classification of quasihomogeneous polynomials in two variables with real coefficients under semialgebraic bi-Lipschitz equivalence in a neighborhood of the origin in ${\mathbb R}^2$. Building on the work of Birbrair, Fernandes, and Panazzolo, our approach is based on reducing the problem to the Lipschitz classification of associated single-variable polynomial functions, called height functions. We establish conditions under which semialgebraic bi-Lipschitz equivalence of quasihomogeneous polynomials corresponds to the Lipschitz equivalence of their height functions. To achieve this, we develop the theory of $β$-transforms and inverse $β$-transforms. As an application, we examine a family of quasihomogeneous polynomials previously used by Henry and Parusiński to show that the bi-Lipschitz equivalence of analytic function germs $({\mathbb R}^2,0)\rightarrow({\mathbb R},0)$ admits continuous moduli. Our results show that semialgebraic bi-Lipschitz equivalence of real quasihomogeneous polynomials in two variables also admits continuous moduli. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_06022 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Semialgebraic Lipschitz equivalence polynomial functions Alvarez, Sergio Algebraic Geometry We investigate the classification of quasihomogeneous polynomials in two variables with real coefficients under semialgebraic bi-Lipschitz equivalence in a neighborhood of the origin in ${\mathbb R}^2$. Building on the work of Birbrair, Fernandes, and Panazzolo, our approach is based on reducing the problem to the Lipschitz classification of associated single-variable polynomial functions, called height functions. We establish conditions under which semialgebraic bi-Lipschitz equivalence of quasihomogeneous polynomials corresponds to the Lipschitz equivalence of their height functions. To achieve this, we develop the theory of $β$-transforms and inverse $β$-transforms. As an application, we examine a family of quasihomogeneous polynomials previously used by Henry and Parusiński to show that the bi-Lipschitz equivalence of analytic function germs $({\mathbb R}^2,0)\rightarrow({\mathbb R},0)$ admits continuous moduli. Our results show that semialgebraic bi-Lipschitz equivalence of real quasihomogeneous polynomials in two variables also admits continuous moduli. |
| title | Semialgebraic Lipschitz equivalence polynomial functions |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2503.06022 |