The Hilbert–Schinzel specialization property
We establish a version “over the ring” of the celebrated Hilbert Irreducibility Theorem. Given finitely many polynomials in variables, with coefficients in , of positive degree in the last variables, we show that if they are irreducible over and satisfy a necessary “Schinzel condition”, then the fir...
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| Veröffentlicht in: | Journal für die reine und angewandte Mathematik 2022-04, Vol.2022 (785), p.55-79 |
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| container_issue | 785 |
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| container_title | Journal für die reine und angewandte Mathematik |
| container_volume | 2022 |
| creator | Bodin, Arnaud Dèbes, Pierre König, Joachim Najib, Salah |
| description | We establish a version “over the ring” of
the celebrated Hilbert Irreducibility Theorem. Given finitely many polynomials in
variables,
with coefficients in
, of positive degree in the last
variables, we show that if they are irreducible over
and satisfy a necessary “Schinzel condition”, then the first
variables can be specialized in a Zariski-dense subset of
in such a way that irreducibility over
is preserved
for the polynomials in the remaining
variables. The Schinzel condition, which comes from the Schinzel Hypothesis, is that, when specializing the first
variables in
, the product of the polynomials should not always be divisible by some common prime number. Our result also improves on a “coprime” version of the Schinzel Hypothesis: under some Schinzel condition, coprime polynomials
assume coprime values.
We prove our results over many other rings than
, e.g. UFDs and Dedekind domains. |
| doi_str_mv | 10.1515/crelle-2021-0083 |
| format | Article |
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the celebrated Hilbert Irreducibility Theorem. Given finitely many polynomials in
variables,
with coefficients in
, of positive degree in the last
variables, we show that if they are irreducible over
and satisfy a necessary “Schinzel condition”, then the first
variables can be specialized in a Zariski-dense subset of
in such a way that irreducibility over
is preserved
for the polynomials in the remaining
variables. The Schinzel condition, which comes from the Schinzel Hypothesis, is that, when specializing the first
variables in
, the product of the polynomials should not always be divisible by some common prime number. Our result also improves on a “coprime” version of the Schinzel Hypothesis: under some Schinzel condition, coprime polynomials
assume coprime values.
We prove our results over many other rings than
, e.g. UFDs and Dedekind domains.</description><identifier>ISSN: 0075-4102</identifier><identifier>EISSN: 1435-5345</identifier><identifier>DOI: 10.1515/crelle-2021-0083</identifier><language>eng</language><publisher>De Gruyter</publisher><subject>Mathematics ; Number Theory</subject><ispartof>Journal für die reine und angewandte Mathematik, 2022-04, Vol.2022 (785), p.55-79</ispartof><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c335t-c7cc9964fec2a9b588b7e8a162193a2f182f9eab1b9140725c83eb1d5de5d74f3</citedby><cites>FETCH-LOGICAL-c335t-c7cc9964fec2a9b588b7e8a162193a2f182f9eab1b9140725c83eb1d5de5d74f3</cites><orcidid>0000-0001-9506-1380</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.degruyter.com/document/doi/10.1515/crelle-2021-0083/pdf$$EPDF$$P50$$Gwalterdegruyter$$H</linktopdf><linktohtml>$$Uhttps://www.degruyter.com/document/doi/10.1515/crelle-2021-0083/html$$EHTML$$P50$$Gwalterdegruyter$$H</linktohtml><link.rule.ids>230,314,776,780,881,27903,27904,66500,68284</link.rule.ids><backlink>$$Uhttps://hal.science/hal-03800676$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Bodin, Arnaud</creatorcontrib><creatorcontrib>Dèbes, Pierre</creatorcontrib><creatorcontrib>König, Joachim</creatorcontrib><creatorcontrib>Najib, Salah</creatorcontrib><title>The Hilbert–Schinzel specialization property</title><title>Journal für die reine und angewandte Mathematik</title><description>We establish a version “over the ring” of
the celebrated Hilbert Irreducibility Theorem. Given finitely many polynomials in
variables,
with coefficients in
, of positive degree in the last
variables, we show that if they are irreducible over
and satisfy a necessary “Schinzel condition”, then the first
variables can be specialized in a Zariski-dense subset of
in such a way that irreducibility over
is preserved
for the polynomials in the remaining
variables. The Schinzel condition, which comes from the Schinzel Hypothesis, is that, when specializing the first
variables in
, the product of the polynomials should not always be divisible by some common prime number. Our result also improves on a “coprime” version of the Schinzel Hypothesis: under some Schinzel condition, coprime polynomials
assume coprime values.
We prove our results over many other rings than
, e.g. UFDs and Dedekind domains.</description><subject>Mathematics</subject><subject>Number Theory</subject><issn>0075-4102</issn><issn>1435-5345</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><recordid>eNp1kLFOwzAQhi0EEqWwM3ZlSLmz49gZqwooUiUGymw5zoWmMk3kpKB04h14Q56EREFsTPfr9H8n3cfYNcIcJcpbF8h7ijhwjAC0OGETjIWMpIjlKZsAKBnFCPycXTTNDgAkKj5h882WZqvSZxTa78-vZ7ct90fys6YmV1pfHm1bVvtZHaq6b3SX7KywvqGr3zllL_d3m-UqWj89PC4X68gJIdvIKefSNIkLctymmdQ6U6QtJhxTYXmBmhcp2QyzFGNQXDotKMNc5iRzFRdiym7Gu1vrTR3KNxs6U9nSrBZrM-xAaIBEJe_Yd2HsulA1TaDiD0AwgxszujGDGzO46ZHFiHxY31LI6TUcuj6YXXUI-_6xf9E-cKWllOIHk5BuSw</recordid><startdate>20220401</startdate><enddate>20220401</enddate><creator>Bodin, Arnaud</creator><creator>Dèbes, Pierre</creator><creator>König, Joachim</creator><creator>Najib, Salah</creator><general>De Gruyter</general><general>Walter de Gruyter</general><scope>AAYXX</scope><scope>CITATION</scope><scope>1XC</scope><scope>VOOES</scope><orcidid>https://orcid.org/0000-0001-9506-1380</orcidid></search><sort><creationdate>20220401</creationdate><title>The Hilbert–Schinzel specialization property</title><author>Bodin, Arnaud ; Dèbes, Pierre ; König, Joachim ; Najib, Salah</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c335t-c7cc9964fec2a9b588b7e8a162193a2f182f9eab1b9140725c83eb1d5de5d74f3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Mathematics</topic><topic>Number Theory</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bodin, Arnaud</creatorcontrib><creatorcontrib>Dèbes, Pierre</creatorcontrib><creatorcontrib>König, Joachim</creatorcontrib><creatorcontrib>Najib, Salah</creatorcontrib><collection>CrossRef</collection><collection>Hyper Article en Ligne (HAL)</collection><collection>Hyper Article en Ligne (HAL) (Open Access)</collection><jtitle>Journal für die reine und angewandte Mathematik</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bodin, Arnaud</au><au>Dèbes, Pierre</au><au>König, Joachim</au><au>Najib, Salah</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The Hilbert–Schinzel specialization property</atitle><jtitle>Journal für die reine und angewandte Mathematik</jtitle><date>2022-04-01</date><risdate>2022</risdate><volume>2022</volume><issue>785</issue><spage>55</spage><epage>79</epage><pages>55-79</pages><issn>0075-4102</issn><eissn>1435-5345</eissn><abstract>We establish a version “over the ring” of
the celebrated Hilbert Irreducibility Theorem. Given finitely many polynomials in
variables,
with coefficients in
, of positive degree in the last
variables, we show that if they are irreducible over
and satisfy a necessary “Schinzel condition”, then the first
variables can be specialized in a Zariski-dense subset of
in such a way that irreducibility over
is preserved
for the polynomials in the remaining
variables. The Schinzel condition, which comes from the Schinzel Hypothesis, is that, when specializing the first
variables in
, the product of the polynomials should not always be divisible by some common prime number. Our result also improves on a “coprime” version of the Schinzel Hypothesis: under some Schinzel condition, coprime polynomials
assume coprime values.
We prove our results over many other rings than
, e.g. UFDs and Dedekind domains.</abstract><pub>De Gruyter</pub><doi>10.1515/crelle-2021-0083</doi><tpages>25</tpages><orcidid>https://orcid.org/0000-0001-9506-1380</orcidid><oa>free_for_read</oa></addata></record> |
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| identifier | ISSN: 0075-4102 |
| ispartof | Journal für die reine und angewandte Mathematik, 2022-04, Vol.2022 (785), p.55-79 |
| issn | 0075-4102 1435-5345 |
| language | eng |
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| source | De Gruyter journals |
| subjects | Mathematics Number Theory |
| title | The Hilbert–Schinzel specialization property |
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