Hilbert's Irreducibility Theorem for Linear Differential Operators
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910375005388800 |
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| author | Feng, Ruyong Guo, Zewang Lu, Wei |
| author_facet | Feng, Ruyong Guo, Zewang Lu, Wei |
| contents | We prove a differential analogue of Hilbert's irreducibility theorem. Let $\mathcal{L}$ be a linear differential operator with coefficients in $C(\mathbb{X})(x)$ that is irreducible over $\overline{C(\mathbb{X})}(x)$, where $\mathbb{X}$ is an irreducible affine algebraic variety over an algebraically closed field $C$ of characteristic zero. We show that the set of $c\in \mathbb{X}(C)$ such that the specialized operator $\mathcal{L}^c$ of $\mathcal{L}$ remains irreducible over $C(x)$ is Zariski dense in $\mathbb{X}(C)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_13228 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hilbert's Irreducibility Theorem for Linear Differential Operators Feng, Ruyong Guo, Zewang Lu, Wei Rings and Algebras Classical Analysis and ODEs 16S32, 68W30 We prove a differential analogue of Hilbert's irreducibility theorem. Let $\mathcal{L}$ be a linear differential operator with coefficients in $C(\mathbb{X})(x)$ that is irreducible over $\overline{C(\mathbb{X})}(x)$, where $\mathbb{X}$ is an irreducible affine algebraic variety over an algebraically closed field $C$ of characteristic zero. We show that the set of $c\in \mathbb{X}(C)$ such that the specialized operator $\mathcal{L}^c$ of $\mathcal{L}$ remains irreducible over $C(x)$ is Zariski dense in $\mathbb{X}(C)$. |
| title | Hilbert's Irreducibility Theorem for Linear Differential Operators |
| topic | Rings and Algebras Classical Analysis and ODEs 16S32, 68W30 |
| url | https://arxiv.org/abs/2403.13228 |