Bruno ideal and the variety of centers for singular germs of vector fields
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917274580942848 |
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| author | Martín-Vega, María Panazzolo, Daniel |
| author_facet | Martín-Vega, María Panazzolo, Daniel |
| contents | Given a logarithmic analytic vector field $\partial$, we consider the formal ideal $B(\partial)$ defined by the collinearity locus of the semi-simple and nilpotent components of~$\partial$. Assuming that the eigenvalues of the linear part of $\partial$ satisfy the so-called Bruno arithmetic condition, we prove that $B(\partial)$ is in fact an analytic ideal. Moreover, $\partial$ is analytically normalizable when restricted to this ideal. As a consequence, the vanishing locus $V$ of $B(\partial)$ is an analytic variety, and the foliation defined by $\partial|_{V}$ is analytically linearizable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13527 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bruno ideal and the variety of centers for singular germs of vector fields Martín-Vega, María Panazzolo, Daniel Dynamical Systems 34C20, 37G05, 37L10, 34M35 Given a logarithmic analytic vector field $\partial$, we consider the formal ideal $B(\partial)$ defined by the collinearity locus of the semi-simple and nilpotent components of~$\partial$. Assuming that the eigenvalues of the linear part of $\partial$ satisfy the so-called Bruno arithmetic condition, we prove that $B(\partial)$ is in fact an analytic ideal. Moreover, $\partial$ is analytically normalizable when restricted to this ideal. As a consequence, the vanishing locus $V$ of $B(\partial)$ is an analytic variety, and the foliation defined by $\partial|_{V}$ is analytically linearizable. |
| title | Bruno ideal and the variety of centers for singular germs of vector fields |
| topic | Dynamical Systems 34C20, 37G05, 37L10, 34M35 |
| url | https://arxiv.org/abs/2602.13527 |