Some irreducibility and indecomposability results for truncated binomial polynomials of small degree
In this paper, we show that the truncated binomial polynomials defined by P n , k ( x ) = ∑ j = 0 k n j x j are irreducible for each k ≤6 and every n ≥ k +2. Under the same assumption n ≥ k +2, we also show that the polynomial P n , k cannot be expressed as a composition P n , k ( x ) = g ( h ( x ))...
Gespeichert in:
| Veröffentlicht in: | Proceedings of the Indian Academy of Sciences. Mathematical sciences 2017-02, Vol.127 (1), p.45-57 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| container_end_page | 57 |
|---|---|
| container_issue | 1 |
| container_start_page | 45 |
| container_title | Proceedings of the Indian Academy of Sciences. Mathematical sciences |
| container_volume | 127 |
| creator | DUBICKAS, ARTŪRAS ŠIURYS, JONAS |
| description | In this paper, we show that the truncated binomial polynomials defined by
P
n
,
k
(
x
)
=
∑
j
=
0
k
n
j
x
j
are irreducible for each
k
≤6 and every
n
≥
k
+2. Under the same assumption
n
≥
k
+2, we also show that the polynomial
P
n
,
k
cannot be expressed as a composition
P
n
,
k
(
x
) =
g
(
h
(
x
)) with
g
∈
ℚ
[
x
]
of degree at least 2 and a quadratic polynomial
h
∈
ℚ
[
x
]
. Finally, we show that for
k
≥2 and
m
,
n
≥
k
+1 the roots of the polynomial
P
m
,
k
cannot be obtained from the roots of
P
n
,
k
, where
m
≠
n
, by a linear map. |
| doi_str_mv | 10.1007/s12044-016-0325-0 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_1880783634</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>1880783634</sourcerecordid><originalsourceid>FETCH-LOGICAL-c316t-c680d82bdcf9f09ade1fbf7f831641a92cac837182d4671a4716e3c30042277f3</originalsourceid><addsrcrecordid>eNp1UMtKBDEQDKLguvoB3gKeo3ltkjnK4gsWPKjnkMljyTIzGZOZw_69WcaDFy_dRXVVNRQAtwTfE4zlQyEUc44wEQgzukH4DKxwIxmSQm3OK6Ybhjjh9BJclXLAmDSciRVwH6n3MObs3WxjG7s4HaEZHIyD8zb1Yyrml82-zN1UYEgZTnkerJm8g20cUh9NB8fUHRdYYAqw9KbroPP77P01uAiV9je_ew2-np8-t69o9_7ytn3cIcuImJAVCjtFW2dDE3BjnCehDTKoeuXENNQaq5gkijouJDFcEuGZZRhzSqUMbA3ultwxp-_Zl0kf0pyH-lITpbBUTDBeVWRR2ZxKyT7oMcfe5KMmWJ_K1EuZupapT2XWsQZ08ZSqHfY-_0n-1_QDKXR4wQ</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>1880783634</pqid></control><display><type>article</type><title>Some irreducibility and indecomposability results for truncated binomial polynomials of small degree</title><source>Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals</source><source>Indian Academy of Sciences</source><source>Springer Nature - Complete Springer Journals</source><creator>DUBICKAS, ARTŪRAS ; ŠIURYS, JONAS</creator><creatorcontrib>DUBICKAS, ARTŪRAS ; ŠIURYS, JONAS</creatorcontrib><description>In this paper, we show that the truncated binomial polynomials defined by
P
n
,
k
(
x
)
=
∑
j
=
0
k
n
j
x
j
are irreducible for each
k
≤6 and every
n
≥
k
+2. Under the same assumption
n
≥
k
+2, we also show that the polynomial
P
n
,
k
cannot be expressed as a composition
P
n
,
k
(
x
) =
g
(
h
(
x
)) with
g
∈
ℚ
[
x
]
of degree at least 2 and a quadratic polynomial
h
∈
ℚ
[
x
]
. Finally, we show that for
k
≥2 and
m
,
n
≥
k
+1 the roots of the polynomial
P
m
,
k
cannot be obtained from the roots of
P
n
,
k
, where
m
≠
n
, by a linear map.</description><identifier>ISSN: 0253-4142</identifier><identifier>EISSN: 0973-7685</identifier><identifier>DOI: 10.1007/s12044-016-0325-0</identifier><language>eng</language><publisher>New Delhi: Springer India</publisher><subject>Mathematics ; Mathematics and Statistics ; Polynomials ; Roots</subject><ispartof>Proceedings of the Indian Academy of Sciences. Mathematical sciences, 2017-02, Vol.127 (1), p.45-57</ispartof><rights>Indian Academy of Sciences 2016</rights><rights>Copyright Springer Science & Business Media 2017</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-c680d82bdcf9f09ade1fbf7f831641a92cac837182d4671a4716e3c30042277f3</citedby><cites>FETCH-LOGICAL-c316t-c680d82bdcf9f09ade1fbf7f831641a92cac837182d4671a4716e3c30042277f3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s12044-016-0325-0$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s12044-016-0325-0$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27903,27904,41467,42536,51297</link.rule.ids></links><search><creatorcontrib>DUBICKAS, ARTŪRAS</creatorcontrib><creatorcontrib>ŠIURYS, JONAS</creatorcontrib><title>Some irreducibility and indecomposability results for truncated binomial polynomials of small degree</title><title>Proceedings of the Indian Academy of Sciences. Mathematical sciences</title><addtitle>Proc Math Sci</addtitle><description>In this paper, we show that the truncated binomial polynomials defined by
P
n
,
k
(
x
)
=
∑
j
=
0
k
n
j
x
j
are irreducible for each
k
≤6 and every
n
≥
k
+2. Under the same assumption
n
≥
k
+2, we also show that the polynomial
P
n
,
k
cannot be expressed as a composition
P
n
,
k
(
x
) =
g
(
h
(
x
)) with
g
∈
ℚ
[
x
]
of degree at least 2 and a quadratic polynomial
h
∈
ℚ
[
x
]
. Finally, we show that for
k
≥2 and
m
,
n
≥
k
+1 the roots of the polynomial
P
m
,
k
cannot be obtained from the roots of
P
n
,
k
, where
m
≠
n
, by a linear map.</description><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Polynomials</subject><subject>Roots</subject><issn>0253-4142</issn><issn>0973-7685</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><recordid>eNp1UMtKBDEQDKLguvoB3gKeo3ltkjnK4gsWPKjnkMljyTIzGZOZw_69WcaDFy_dRXVVNRQAtwTfE4zlQyEUc44wEQgzukH4DKxwIxmSQm3OK6Ybhjjh9BJclXLAmDSciRVwH6n3MObs3WxjG7s4HaEZHIyD8zb1Yyrml82-zN1UYEgZTnkerJm8g20cUh9NB8fUHRdYYAqw9KbroPP77P01uAiV9je_ew2-np8-t69o9_7ytn3cIcuImJAVCjtFW2dDE3BjnCehDTKoeuXENNQaq5gkijouJDFcEuGZZRhzSqUMbA3ultwxp-_Zl0kf0pyH-lITpbBUTDBeVWRR2ZxKyT7oMcfe5KMmWJ_K1EuZupapT2XWsQZ08ZSqHfY-_0n-1_QDKXR4wQ</recordid><startdate>20170201</startdate><enddate>20170201</enddate><creator>DUBICKAS, ARTŪRAS</creator><creator>ŠIURYS, JONAS</creator><general>Springer India</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20170201</creationdate><title>Some irreducibility and indecomposability results for truncated binomial polynomials of small degree</title><author>DUBICKAS, ARTŪRAS ; ŠIURYS, JONAS</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-c680d82bdcf9f09ade1fbf7f831641a92cac837182d4671a4716e3c30042277f3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Polynomials</topic><topic>Roots</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>DUBICKAS, ARTŪRAS</creatorcontrib><creatorcontrib>ŠIURYS, JONAS</creatorcontrib><collection>CrossRef</collection><jtitle>Proceedings of the Indian Academy of Sciences. Mathematical sciences</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>DUBICKAS, ARTŪRAS</au><au>ŠIURYS, JONAS</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Some irreducibility and indecomposability results for truncated binomial polynomials of small degree</atitle><jtitle>Proceedings of the Indian Academy of Sciences. Mathematical sciences</jtitle><stitle>Proc Math Sci</stitle><date>2017-02-01</date><risdate>2017</risdate><volume>127</volume><issue>1</issue><spage>45</spage><epage>57</epage><pages>45-57</pages><issn>0253-4142</issn><eissn>0973-7685</eissn><abstract>In this paper, we show that the truncated binomial polynomials defined by
P
n
,
k
(
x
)
=
∑
j
=
0
k
n
j
x
j
are irreducible for each
k
≤6 and every
n
≥
k
+2. Under the same assumption
n
≥
k
+2, we also show that the polynomial
P
n
,
k
cannot be expressed as a composition
P
n
,
k
(
x
) =
g
(
h
(
x
)) with
g
∈
ℚ
[
x
]
of degree at least 2 and a quadratic polynomial
h
∈
ℚ
[
x
]
. Finally, we show that for
k
≥2 and
m
,
n
≥
k
+1 the roots of the polynomial
P
m
,
k
cannot be obtained from the roots of
P
n
,
k
, where
m
≠
n
, by a linear map.</abstract><cop>New Delhi</cop><pub>Springer India</pub><doi>10.1007/s12044-016-0325-0</doi><tpages>13</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0253-4142 |
| ispartof | Proceedings of the Indian Academy of Sciences. Mathematical sciences, 2017-02, Vol.127 (1), p.45-57 |
| issn | 0253-4142 0973-7685 |
| language | eng |
| recordid | cdi_proquest_journals_1880783634 |
| source | Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals; Indian Academy of Sciences; Springer Nature - Complete Springer Journals |
| subjects | Mathematics Mathematics and Statistics Polynomials Roots |
| title | Some irreducibility and indecomposability results for truncated binomial polynomials of small degree |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-24T04%3A23%3A34IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_cross&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Some%20irreducibility%20and%20indecomposability%20results%20for%20truncated%20binomial%20polynomials%20of%20small%20degree&rft.jtitle=Proceedings%20of%20the%20Indian%20Academy%20of%20Sciences.%20Mathematical%20sciences&rft.au=DUBICKAS,%20ART%C5%AARAS&rft.date=2017-02-01&rft.volume=127&rft.issue=1&rft.spage=45&rft.epage=57&rft.pages=45-57&rft.issn=0253-4142&rft.eissn=0973-7685&rft_id=info:doi/10.1007/s12044-016-0325-0&rft_dat=%3Cproquest_cross%3E1880783634%3C/proquest_cross%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=1880783634&rft_id=info:pmid/&rfr_iscdi=true |