On the image of polynomials evaluated on incidence algebras: a counter-example and a solution

In this paper, we investigate the subset obtained by evaluations of a fixed multilinear polynomial on a given algebra. We provide an example of a multilinear polynomial, whose image is not a vector subspace; namely, the product of two commutators need not to be a subspace whenever evaluated on certa...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2019-04
Main Authors: Santulo, Ednei A, Yasumura, Felipe Y
Format: Article
Language:English
Subjects:
Algebra
Chief executive officers
Commutators
Fields (mathematics)
Incidence
Matrices (mathematics)
Polynomials
Set theory
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
cited_by
cites
container_end_page
container_issue
container_start_page
container_title arXiv.org
container_volume
creator Santulo, Ednei A
Yasumura, Felipe Y
description In this paper, we investigate the subset obtained by evaluations of a fixed multilinear polynomial on a given algebra. We provide an example of a multilinear polynomial, whose image is not a vector subspace; namely, the product of two commutators need not to be a subspace whenever evaluated on certain subalgebras of upper triangular matrices (the so-called incidence algebras). In the last part of the paper, given that the field is infinite, we reduce the problem of the description of the image of a polynomial evaluated on an incidence algebra to the study of evaluations of a certain family of polynomials on its Jacobson radical. In particular, we are able to describe the image of multilinear polynomials evaluated on the algebra of upper triangular matrices.
format article
fullrecord <record><control><sourceid>proquest</sourceid><recordid>TN_cdi_proquest_journals_2184848088</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2184848088</sourcerecordid><originalsourceid>FETCH-proquest_journals_21848480883</originalsourceid><addsrcrecordid>eNqNjMEKwjAQRIMgWLT_sOC5UNNWi1dRvHnxKiW225qS7tYmEf17c_ADZA4P5g0zE5HMsk1S5lIuRGxtn6ap3O5kUWSRuF0I3ANBD6pD4BZGNh_iQStjAV_KeOWwASbQVOsGqUZQpsP7pOweFNTsyeGU4FsNowmOmtBaNt5pppWYt-EI4x-XYn06Xg_nZJz46dG6qmc_UVCV3JR5SFqW2X-rL-_XQ7U</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2184848088</pqid></control><display><type>article</type><title>On the image of polynomials evaluated on incidence algebras: a counter-example and a solution</title><source>Publicly Available Content Database</source><creator>Santulo, Ednei A ; Yasumura, Felipe Y</creator><creatorcontrib>Santulo, Ednei A ; Yasumura, Felipe Y</creatorcontrib><description>In this paper, we investigate the subset obtained by evaluations of a fixed multilinear polynomial on a given algebra. We provide an example of a multilinear polynomial, whose image is not a vector subspace; namely, the product of two commutators need not to be a subspace whenever evaluated on certain subalgebras of upper triangular matrices (the so-called incidence algebras). In the last part of the paper, given that the field is infinite, we reduce the problem of the description of the image of a polynomial evaluated on an incidence algebra to the study of evaluations of a certain family of polynomials on its Jacobson radical. In particular, we are able to describe the image of multilinear polynomials evaluated on the algebra of upper triangular matrices.</description><identifier>EISSN: 2331-8422</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Algebra ; Chief executive officers ; Commutators ; Fields (mathematics) ; Incidence ; Matrices (mathematics) ; Polynomials ; Set theory</subject><ispartof>arXiv.org, 2019-04</ispartof><rights>2019. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.proquest.com/docview/2184848088?pq-origsite=primo$$EHTML$$P50$$Gproquest$$Hfree_for_read</linktohtml><link.rule.ids>780,784,25751,37010,44588</link.rule.ids></links><search><creatorcontrib>Santulo, Ednei A</creatorcontrib><creatorcontrib>Yasumura, Felipe Y</creatorcontrib><title>On the image of polynomials evaluated on incidence algebras: a counter-example and a solution</title><title>arXiv.org</title><description>In this paper, we investigate the subset obtained by evaluations of a fixed multilinear polynomial on a given algebra. We provide an example of a multilinear polynomial, whose image is not a vector subspace; namely, the product of two commutators need not to be a subspace whenever evaluated on certain subalgebras of upper triangular matrices (the so-called incidence algebras). In the last part of the paper, given that the field is infinite, we reduce the problem of the description of the image of a polynomial evaluated on an incidence algebra to the study of evaluations of a certain family of polynomials on its Jacobson radical. In particular, we are able to describe the image of multilinear polynomials evaluated on the algebra of upper triangular matrices.</description><subject>Algebra</subject><subject>Chief executive officers</subject><subject>Commutators</subject><subject>Fields (mathematics)</subject><subject>Incidence</subject><subject>Matrices (mathematics)</subject><subject>Polynomials</subject><subject>Set theory</subject><issn>2331-8422</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><sourceid>PIMPY</sourceid><recordid>eNqNjMEKwjAQRIMgWLT_sOC5UNNWi1dRvHnxKiW225qS7tYmEf17c_ADZA4P5g0zE5HMsk1S5lIuRGxtn6ap3O5kUWSRuF0I3ANBD6pD4BZGNh_iQStjAV_KeOWwASbQVOsGqUZQpsP7pOweFNTsyeGU4FsNowmOmtBaNt5pppWYt-EI4x-XYn06Xg_nZJz46dG6qmc_UVCV3JR5SFqW2X-rL-_XQ7U</recordid><startdate>20190420</startdate><enddate>20190420</enddate><creator>Santulo, Ednei A</creator><creator>Yasumura, Felipe Y</creator><general>Cornell University Library, arXiv.org</general><scope>8FE</scope><scope>8FG</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>HCIFZ</scope><scope>L6V</scope><scope>M7S</scope><scope>PIMPY</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope></search><sort><creationdate>20190420</creationdate><title>On the image of polynomials evaluated on incidence algebras: a counter-example and a solution</title><author>Santulo, Ednei A ; Yasumura, Felipe Y</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-proquest_journals_21848480883</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Algebra</topic><topic>Chief executive officers</topic><topic>Commutators</topic><topic>Fields (mathematics)</topic><topic>Incidence</topic><topic>Matrices (mathematics)</topic><topic>Polynomials</topic><topic>Set theory</topic><toplevel>online_resources</toplevel><creatorcontrib>Santulo, Ednei A</creatorcontrib><creatorcontrib>Yasumura, Felipe Y</creatorcontrib><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>Materials Science &amp; Engineering Collection</collection><collection>ProQuest Central (Alumni)</collection><collection>ProQuest Central</collection><collection>ProQuest Central Essentials</collection><collection>AUTh Library subscriptions: ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Engineering Collection</collection><collection>Engineering Database</collection><collection>Publicly Available Content Database</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering Collection</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Santulo, Ednei A</au><au>Yasumura, Felipe Y</au><format>book</format><genre>document</genre><ristype>GEN</ristype><atitle>On the image of polynomials evaluated on incidence algebras: a counter-example and a solution</atitle><jtitle>arXiv.org</jtitle><date>2019-04-20</date><risdate>2019</risdate><eissn>2331-8422</eissn><abstract>In this paper, we investigate the subset obtained by evaluations of a fixed multilinear polynomial on a given algebra. We provide an example of a multilinear polynomial, whose image is not a vector subspace; namely, the product of two commutators need not to be a subspace whenever evaluated on certain subalgebras of upper triangular matrices (the so-called incidence algebras). In the last part of the paper, given that the field is infinite, we reduce the problem of the description of the image of a polynomial evaluated on an incidence algebra to the study of evaluations of a certain family of polynomials on its Jacobson radical. In particular, we are able to describe the image of multilinear polynomials evaluated on the algebra of upper triangular matrices.</abstract><cop>Ithaca</cop><pub>Cornell University Library, arXiv.org</pub><oa>free_for_read</oa></addata></record>
fulltext fulltext
identifier EISSN: 2331-8422
ispartof arXiv.org, 2019-04
issn 2331-8422
language eng
recordid cdi_proquest_journals_2184848088
source Publicly Available Content Database
subjects Algebra
Chief executive officers
Commutators
Fields (mathematics)
Incidence
Matrices (mathematics)
Polynomials
Set theory
title On the image of polynomials evaluated on incidence algebras: a counter-example and a solution
url http://sfxeu10.hosted.exlibrisgroup.com/loughborough?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-09T19%3A08%3A09IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest&rft_val_fmt=info:ofi/fmt:kev:mtx:book&rft.genre=document&rft.atitle=On%20the%20image%20of%20polynomials%20evaluated%20on%20incidence%20algebras:%20a%20counter-example%20and%20a%20solution&rft.jtitle=arXiv.org&rft.au=Santulo,%20Ednei%20A&rft.date=2019-04-20&rft.eissn=2331-8422&rft_id=info:doi/&rft_dat=%3Cproquest%3E2184848088%3C/proquest%3E%3Cgrp_id%3Ecdi_FETCH-proquest_journals_21848480883%3C/grp_id%3E%3Coa%3E%3C/oa%3E%3Curl%3E%3C/url%3E&rft_id=info:oai/&rft_pqid=2184848088&rft_id=info:pmid/&rfr_iscdi=true