Hilbert's Irreducibility Theorem for Linear Differential Operators
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 algebra...
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| creator | Feng, Ruyong Guo, Zewang Lu, Wei |
| description | 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
$\mathbb{X}(C)$. |
| doi_str_mv | 10.48550/arxiv.2403.13228 |
| format | Article |
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$\mathcal{L}$ 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
$\mathbb{X}(C)$.</description><identifier>DOI: 10.48550/arxiv.2403.13228</identifier><language>eng</language><subject>Mathematics - Classical Analysis and ODEs ; Mathematics - Rings and Algebras</subject><creationdate>2024-03</creationdate><rights>http://creativecommons.org/licenses/by/4.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,776,881</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/2403.13228$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.2403.13228$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Feng, Ruyong</creatorcontrib><creatorcontrib>Guo, Zewang</creatorcontrib><creatorcontrib>Lu, Wei</creatorcontrib><title>Hilbert's Irreducibility Theorem for Linear Differential Operators</title><description>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
$\mathbb{X}(C)$.</description><subject>Mathematics - Classical Analysis and ODEs</subject><subject>Mathematics - Rings and Algebras</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotz71OwzAUQGEvDKjlAZjw1inBuY4dZ4Ty00qRumSPru1rYSltqpuA6NsjCtPZjvQJcV-psnbGqEfk7_xVQq10WWkAdyued3n0xMtmlntmip8h-zzm5SL7D5qYjjJNLLt8ImT5klMiptOScZSHMzEuE89rcZNwnOnuvyvRv732213RHd7326euQNu4QuvKWqcUNIBtVKYNESBC1B4qp72nZAJaXduIZJNyoalsDcGS8S1QAr0SD3_bK2I4cz4iX4ZfzHDF6B-rJERu</recordid><startdate>20240319</startdate><enddate>20240319</enddate><creator>Feng, Ruyong</creator><creator>Guo, Zewang</creator><creator>Lu, Wei</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20240319</creationdate><title>Hilbert's Irreducibility Theorem for Linear Differential Operators</title><author>Feng, Ruyong ; Guo, Zewang ; Lu, Wei</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a678-33166800272a9d059cd22d2d3b2183bbef5ca6346dae6f08c71642c6e5b92ef23</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Mathematics - Classical Analysis and ODEs</topic><topic>Mathematics - Rings and Algebras</topic><toplevel>online_resources</toplevel><creatorcontrib>Feng, Ruyong</creatorcontrib><creatorcontrib>Guo, Zewang</creatorcontrib><creatorcontrib>Lu, Wei</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Feng, Ruyong</au><au>Guo, Zewang</au><au>Lu, Wei</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Hilbert's Irreducibility Theorem for Linear Differential Operators</atitle><date>2024-03-19</date><risdate>2024</risdate><abstract>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
$\mathbb{X}(C)$.</abstract><doi>10.48550/arxiv.2403.13228</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Classical Analysis and ODEs Mathematics - Rings and Algebras |
| title | Hilbert's Irreducibility Theorem for Linear Differential Operators |
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