Bivariate Polynomial Matrix and Smith Form Article Swipe

PDF

YOU? · · 2024 · Open Access · · DOI: https://doi.org/10.3390/math12060815

Matrix equivalence plays a pivotal role in multidimensional systems, which are typically represented by multivariate polynomial matrices. The Smith form of matrices is one of the important research topics in polynomial matrices. This article mainly investigates the Smith forms of several types of bivariate polynomial matrices and has successfully derived several necessary and sufficient conditions for matrix equivalence.

Related Topics

Mathematics

Mathematical Analysis

Statistics

Composite Material

Materials Science

Concepts

Polynomial matrix Matrix polynomial Bivariate analysis Mathematics Equivalence (formal languages) Stable polynomial Matrix (chemical analysis) Polynomial Characteristic polynomial Algebra over a field Companion matrix Pure mathematics Alternating polynomial Mathematical analysis Statistics Composite material Materials science

Metadata

Type: article
Language: en
Landing Page: https://doi.org/10.3390/math12060815
PDF: https://www.mdpi.com/2227-7390/12/6/815/pdf?version=1710075190
OA Status: gold
References: 24
Related Works: 10
OpenAlex ID: https://openalex.org/W4392647116

All OpenAlex metadata

Raw OpenAlex JSON

OpenAlex ID: https://openalex.org/W4392647116

Canonical identifier for this work in OpenAlex
DOI: https://doi.org/10.3390/math12060815

Digital Object Identifier
Title: Bivariate Polynomial Matrix and Smith Form

Work title
Type: article

OpenAlex work type
Language: en

Primary language
Publication year: 2024

Year of publication
Publication date: 2024-03-10

Full publication date if available
Authors: Licui Zheng, Tao Wu, Jinwang Liu

List of authors in order
Landing page: https://doi.org/10.3390/math12060815

Publisher landing page
PDF URL: https://www.mdpi.com/2227-7390/12/6/815/pdf?version=1710075190

Direct link to full text PDF
Open access: Yes

Whether a free full text is available
OA status: gold

Open access status per OpenAlex
OA URL: https://www.mdpi.com/2227-7390/12/6/815/pdf?version=1710075190

Direct OA link when available
Concepts: Polynomial matrix, Matrix polynomial, Bivariate analysis, Mathematics, Equivalence (formal languages), Stable polynomial, Matrix (chemical analysis), Polynomial, Characteristic polynomial, Algebra over a field, Companion matrix, Pure mathematics, Alternating polynomial, Mathematical analysis, Statistics, Composite material, Materials science

Top concepts (fields/topics) attached by OpenAlex
Cited by: 0

Total citation count in OpenAlex
References (count): 24

Number of works referenced by this work
Related works (count): 10

Other works algorithmically related by OpenAlex

Full payload

id	https://openalex.org/W4392647116
doi	https://doi.org/10.3390/math12060815
ids.doi	https://doi.org/10.3390/math12060815
ids.openalex	https://openalex.org/W4392647116
fwci	0.0
type	article
title	Bivariate Polynomial Matrix and Smith Form
awards[0].id	https://openalex.org/G6455544819
awards[0].funder_id	https://openalex.org/F4320321001
awards[0].display_name
awards[0].funder_award_id	12201204
awards[0].funder_display_name	National Natural Science Foundation of China
biblio.issue	6
biblio.volume	12
biblio.last_page	815
biblio.first_page	815
topics[0].id	https://openalex.org/T11435
topics[0].field.id	https://openalex.org/fields/17
topics[0].field.display_name	Computer Science
topics[0].score	0.9961000084877014
topics[0].domain.id	https://openalex.org/domains/3
topics[0].domain.display_name	Physical Sciences
topics[0].subfield.id	https://openalex.org/subfields/1703
topics[0].subfield.display_name	Computational Theory and Mathematics
topics[0].display_name	Polynomial and algebraic computation
topics[1].id	https://openalex.org/T13935
topics[1].field.id	https://openalex.org/fields/17
topics[1].field.display_name	Computer Science
topics[1].score	0.9939000010490417
topics[1].domain.id	https://openalex.org/domains/3
topics[1].domain.display_name	Physical Sciences
topics[1].subfield.id	https://openalex.org/subfields/1702
topics[1].subfield.display_name	Artificial Intelligence
topics[1].display_name	Mathematical Control Systems and Analysis
topics[2].id	https://openalex.org/T12396
topics[2].field.id	https://openalex.org/fields/26
topics[2].field.display_name	Mathematics
topics[2].score	0.9829999804496765
topics[2].domain.id	https://openalex.org/domains/3
topics[2].domain.display_name	Physical Sciences
topics[2].subfield.id	https://openalex.org/subfields/2608
topics[2].subfield.display_name	Geometry and Topology
topics[2].display_name	Advanced Differential Equations and Dynamical Systems
funders[0].id	https://openalex.org/F4320321001
funders[0].ror	https://ror.org/01h0zpd94
funders[0].display_name	National Natural Science Foundation of China
is_xpac	False
apc_list.value	1800
apc_list.currency	CHF
apc_list.value_usd	1949
apc_paid.value	1800
apc_paid.currency	CHF
apc_paid.value_usd	1949
concepts[0].id	https://openalex.org/C126352355
concepts[0].level	4
concepts[0].score	0.8473303318023682
concepts[0].wikidata	https://www.wikidata.org/wiki/Q2562273
concepts[0].display_name	Polynomial matrix
concepts[1].id	https://openalex.org/C101044782
concepts[1].level	3
concepts[1].score	0.7935795783996582
concepts[1].wikidata	https://www.wikidata.org/wiki/Q6787887
concepts[1].display_name	Matrix polynomial
concepts[2].id	https://openalex.org/C64341305
concepts[2].level	2
concepts[2].score	0.6953083276748657
concepts[2].wikidata	https://www.wikidata.org/wiki/Q4919225
concepts[2].display_name	Bivariate analysis
concepts[3].id	https://openalex.org/C33923547
concepts[3].level	0
concepts[3].score	0.6337354779243469
concepts[3].wikidata	https://www.wikidata.org/wiki/Q395
concepts[3].display_name	Mathematics
concepts[4].id	https://openalex.org/C2780069185
concepts[4].level	2
concepts[4].score	0.59701007604599
concepts[4].wikidata	https://www.wikidata.org/wiki/Q7977945
concepts[4].display_name	Equivalence (formal languages)
concepts[5].id	https://openalex.org/C45025165
concepts[5].level	5
concepts[5].score	0.5946025252342224
concepts[5].wikidata	https://www.wikidata.org/wiki/Q13424738
concepts[5].display_name	Stable polynomial
concepts[6].id	https://openalex.org/C106487976
concepts[6].level	2
concepts[6].score	0.5550282597541809
concepts[6].wikidata	https://www.wikidata.org/wiki/Q685816
concepts[6].display_name	Matrix (chemical analysis)
concepts[7].id	https://openalex.org/C90119067
concepts[7].level	2
concepts[7].score	0.5513864159584045
concepts[7].wikidata	https://www.wikidata.org/wiki/Q43260
concepts[7].display_name	Polynomial
concepts[8].id	https://openalex.org/C42321380
concepts[8].level	3
concepts[8].score	0.46627146005630493
concepts[8].wikidata	https://www.wikidata.org/wiki/Q849705
concepts[8].display_name	Characteristic polynomial
concepts[9].id	https://openalex.org/C136119220
concepts[9].level	2
concepts[9].score	0.44180893898010254
concepts[9].wikidata	https://www.wikidata.org/wiki/Q1000660
concepts[9].display_name	Algebra over a field
concepts[10].id	https://openalex.org/C141495983
concepts[10].level	5
concepts[10].score	0.4339371919631958
concepts[10].wikidata	https://www.wikidata.org/wiki/Q814361
concepts[10].display_name	Companion matrix
concepts[11].id	https://openalex.org/C202444582
concepts[11].level	1
concepts[11].score	0.3998015820980072
concepts[11].wikidata	https://www.wikidata.org/wiki/Q837863
concepts[11].display_name	Pure mathematics
concepts[12].id	https://openalex.org/C170412648
concepts[12].level	4
concepts[12].score	0.30535516142845154
concepts[12].wikidata	https://www.wikidata.org/wiki/Q4736415
concepts[12].display_name	Alternating polynomial
concepts[13].id	https://openalex.org/C134306372
concepts[13].level	1
concepts[13].score	0.09390014410018921
concepts[13].wikidata	https://www.wikidata.org/wiki/Q7754
concepts[13].display_name	Mathematical analysis
concepts[14].id	https://openalex.org/C105795698
concepts[14].level	1
concepts[14].score	0.08332830667495728
concepts[14].wikidata	https://www.wikidata.org/wiki/Q12483
concepts[14].display_name	Statistics
concepts[15].id	https://openalex.org/C159985019
concepts[15].level	1
concepts[15].score	0.0
concepts[15].wikidata	https://www.wikidata.org/wiki/Q181790
concepts[15].display_name	Composite material
concepts[16].id	https://openalex.org/C192562407
concepts[16].level	0
concepts[16].score	0.0
concepts[16].wikidata	https://www.wikidata.org/wiki/Q228736
concepts[16].display_name	Materials science
keywords[0].id	https://openalex.org/keywords/polynomial-matrix
keywords[0].score	0.8473303318023682
keywords[0].display_name	Polynomial matrix
keywords[1].id	https://openalex.org/keywords/matrix-polynomial
keywords[1].score	0.7935795783996582
keywords[1].display_name	Matrix polynomial
keywords[2].id	https://openalex.org/keywords/bivariate-analysis
keywords[2].score	0.6953083276748657
keywords[2].display_name	Bivariate analysis
keywords[3].id	https://openalex.org/keywords/mathematics
keywords[3].score	0.6337354779243469
keywords[3].display_name	Mathematics
keywords[4].id	https://openalex.org/keywords/equivalence
keywords[4].score	0.59701007604599
keywords[4].display_name	Equivalence (formal languages)
keywords[5].id	https://openalex.org/keywords/stable-polynomial
keywords[5].score	0.5946025252342224
keywords[5].display_name	Stable polynomial
keywords[6].id	https://openalex.org/keywords/matrix
keywords[6].score	0.5550282597541809
keywords[6].display_name	Matrix (chemical analysis)
keywords[7].id	https://openalex.org/keywords/polynomial
keywords[7].score	0.5513864159584045
keywords[7].display_name	Polynomial
keywords[8].id	https://openalex.org/keywords/characteristic-polynomial
keywords[8].score	0.46627146005630493
keywords[8].display_name	Characteristic polynomial
keywords[9].id	https://openalex.org/keywords/algebra-over-a-field
keywords[9].score	0.44180893898010254
keywords[9].display_name	Algebra over a field
keywords[10].id	https://openalex.org/keywords/companion-matrix
keywords[10].score	0.4339371919631958
keywords[10].display_name	Companion matrix
keywords[11].id	https://openalex.org/keywords/pure-mathematics
keywords[11].score	0.3998015820980072
keywords[11].display_name	Pure mathematics
keywords[12].id	https://openalex.org/keywords/alternating-polynomial
keywords[12].score	0.30535516142845154
keywords[12].display_name	Alternating polynomial
keywords[13].id	https://openalex.org/keywords/mathematical-analysis
keywords[13].score	0.09390014410018921
keywords[13].display_name	Mathematical analysis
keywords[14].id	https://openalex.org/keywords/statistics
keywords[14].score	0.08332830667495728
keywords[14].display_name	Statistics
language	en
locations[0].id	doi:10.3390/math12060815
locations[0].is_oa	True
locations[0].source.id	https://openalex.org/S4210192031
locations[0].source.issn	2227-7390
locations[0].source.type	journal
locations[0].source.is_oa	True
locations[0].source.issn_l	2227-7390
locations[0].source.is_core	True
locations[0].source.is_in_doaj	True
locations[0].source.display_name	Mathematics
locations[0].source.host_organization	https://openalex.org/P4310310987
locations[0].source.host_organization_name	Multidisciplinary Digital Publishing Institute
locations[0].source.host_organization_lineage	https://openalex.org/P4310310987
locations[0].source.host_organization_lineage_names	Multidisciplinary Digital Publishing Institute
locations[0].license	cc-by
locations[0].pdf_url	https://www.mdpi.com/2227-7390/12/6/815/pdf?version=1710075190
locations[0].version	publishedVersion
locations[0].raw_type	journal-article
locations[0].license_id	https://openalex.org/licenses/cc-by
locations[0].is_accepted	True
locations[0].is_published	True
locations[0].raw_source_name	Mathematics
locations[0].landing_page_url	https://doi.org/10.3390/math12060815
locations[1].id	pmh:oai:doaj.org/article:bd2306d9dafe425e8f1300191ba62790
locations[1].is_oa	False
locations[1].source.id	https://openalex.org/S4306401280
locations[1].source.issn
locations[1].source.type	repository
locations[1].source.is_oa	False
locations[1].source.issn_l
locations[1].source.is_core	False
locations[1].source.is_in_doaj	False
locations[1].source.display_name	DOAJ (DOAJ: Directory of Open Access Journals)
locations[1].source.host_organization
locations[1].source.host_organization_name
locations[1].license
locations[1].pdf_url
locations[1].version	submittedVersion
locations[1].raw_type	article
locations[1].license_id
locations[1].is_accepted	False
locations[1].is_published	False
locations[1].raw_source_name	Mathematics, Vol 12, Iss 6, p 815 (2024)
locations[1].landing_page_url	https://doaj.org/article/bd2306d9dafe425e8f1300191ba62790
indexed_in	crossref, doaj
authorships[0].author.id	https://openalex.org/A5005298522
authorships[0].author.orcid
authorships[0].author.display_name	Licui Zheng
authorships[0].countries	CN
authorships[0].affiliations[0].institution_ids	https://openalex.org/I121296143
authorships[0].affiliations[0].raw_affiliation_string	Department of Mathematics and Computing Sciences, Hunan University of Science and Technology, Xiangtan 411201, China
authorships[0].institutions[0].id	https://openalex.org/I121296143
authorships[0].institutions[0].ror	https://ror.org/02m9vrb24
authorships[0].institutions[0].type	education
authorships[0].institutions[0].lineage	https://openalex.org/I121296143
authorships[0].institutions[0].country_code	CN
authorships[0].institutions[0].display_name	Hunan University of Science and Technology
authorships[0].author_position	first
authorships[0].raw_author_name	Licui Zheng
authorships[0].is_corresponding	False
authorships[0].raw_affiliation_strings	Department of Mathematics and Computing Sciences, Hunan University of Science and Technology, Xiangtan 411201, China
authorships[1].author.id	https://openalex.org/A5026224959
authorships[1].author.orcid	https://orcid.org/0000-0003-3822-6366
authorships[1].author.display_name	Tao Wu
authorships[1].countries	CN
authorships[1].affiliations[0].institution_ids	https://openalex.org/I121296143
authorships[1].affiliations[0].raw_affiliation_string	Department of Mathematics and Computing Sciences, Hunan University of Science and Technology, Xiangtan 411201, China
authorships[1].institutions[0].id	https://openalex.org/I121296143
authorships[1].institutions[0].ror	https://ror.org/02m9vrb24
authorships[1].institutions[0].type	education
authorships[1].institutions[0].lineage	https://openalex.org/I121296143
authorships[1].institutions[0].country_code	CN
authorships[1].institutions[0].display_name	Hunan University of Science and Technology
authorships[1].author_position	middle
authorships[1].raw_author_name	Tao Wu
authorships[1].is_corresponding	False
authorships[1].raw_affiliation_strings	Department of Mathematics and Computing Sciences, Hunan University of Science and Technology, Xiangtan 411201, China
authorships[2].author.id	https://openalex.org/A5103228907
authorships[2].author.orcid
authorships[2].author.display_name	Jinwang Liu
authorships[2].countries	CN
authorships[2].affiliations[0].institution_ids	https://openalex.org/I121296143
authorships[2].affiliations[0].raw_affiliation_string	Department of Mathematics and Computing Sciences, Hunan University of Science and Technology, Xiangtan 411201, China
authorships[2].institutions[0].id	https://openalex.org/I121296143
authorships[2].institutions[0].ror	https://ror.org/02m9vrb24
authorships[2].institutions[0].type	education
authorships[2].institutions[0].lineage	https://openalex.org/I121296143
authorships[2].institutions[0].country_code	CN
authorships[2].institutions[0].display_name	Hunan University of Science and Technology
authorships[2].author_position	last
authorships[2].raw_author_name	Jinwang Liu
authorships[2].is_corresponding	True
authorships[2].raw_affiliation_strings	Department of Mathematics and Computing Sciences, Hunan University of Science and Technology, Xiangtan 411201, China
has_content.pdf	True
has_content.grobid_xml	True
is_paratext	False
open_access.is_oa	True
open_access.oa_url	https://www.mdpi.com/2227-7390/12/6/815/pdf?version=1710075190
open_access.oa_status	gold
open_access.any_repository_has_fulltext	False
created_date	2025-10-10T00:00:00
display_name	Bivariate Polynomial Matrix and Smith Form
has_fulltext	True
is_retracted	False
updated_date	2025-11-06T03:46:38.306776
primary_topic.id	https://openalex.org/T11435
primary_topic.field.id	https://openalex.org/fields/17
primary_topic.field.display_name	Computer Science
primary_topic.score	0.9961000084877014
primary_topic.domain.id	https://openalex.org/domains/3
primary_topic.domain.display_name	Physical Sciences
primary_topic.subfield.id	https://openalex.org/subfields/1703
primary_topic.subfield.display_name	Computational Theory and Mathematics
primary_topic.display_name	Polynomial and algebraic computation
related_works	https://openalex.org/W2905919007, https://openalex.org/W2023818677, https://openalex.org/W2099412787, https://openalex.org/W1543706171, https://openalex.org/W2040308058, https://openalex.org/W2140446896, https://openalex.org/W2152434155, https://openalex.org/W2062371205, https://openalex.org/W2386881760, https://openalex.org/W2181814365
cited_by_count	0
locations_count	2
best_oa_location.id	doi:10.3390/math12060815
best_oa_location.is_oa	True
best_oa_location.source.id	https://openalex.org/S4210192031
best_oa_location.source.issn	2227-7390
best_oa_location.source.type	journal
best_oa_location.source.is_oa	True
best_oa_location.source.issn_l	2227-7390
best_oa_location.source.is_core	True
best_oa_location.source.is_in_doaj	True
best_oa_location.source.display_name	Mathematics
best_oa_location.source.host_organization	https://openalex.org/P4310310987
best_oa_location.source.host_organization_name	Multidisciplinary Digital Publishing Institute
best_oa_location.source.host_organization_lineage	https://openalex.org/P4310310987
best_oa_location.source.host_organization_lineage_names	Multidisciplinary Digital Publishing Institute
best_oa_location.license	cc-by
best_oa_location.pdf_url	https://www.mdpi.com/2227-7390/12/6/815/pdf?version=1710075190
best_oa_location.version	publishedVersion
best_oa_location.raw_type	journal-article
best_oa_location.license_id	https://openalex.org/licenses/cc-by
best_oa_location.is_accepted	True
best_oa_location.is_published	True
best_oa_location.raw_source_name	Mathematics
best_oa_location.landing_page_url	https://doi.org/10.3390/math12060815
primary_location.id	doi:10.3390/math12060815
primary_location.is_oa	True
primary_location.source.id	https://openalex.org/S4210192031
primary_location.source.issn	2227-7390
primary_location.source.type	journal
primary_location.source.is_oa	True
primary_location.source.issn_l	2227-7390
primary_location.source.is_core	True
primary_location.source.is_in_doaj	True
primary_location.source.display_name	Mathematics
primary_location.source.host_organization	https://openalex.org/P4310310987
primary_location.source.host_organization_name	Multidisciplinary Digital Publishing Institute
primary_location.source.host_organization_lineage	https://openalex.org/P4310310987
primary_location.source.host_organization_lineage_names	Multidisciplinary Digital Publishing Institute
primary_location.license	cc-by
primary_location.pdf_url	https://www.mdpi.com/2227-7390/12/6/815/pdf?version=1710075190
primary_location.version	publishedVersion
primary_location.raw_type	journal-article
primary_location.license_id	https://openalex.org/licenses/cc-by
primary_location.is_accepted	True
primary_location.is_published	True
primary_location.raw_source_name	Mathematics
primary_location.landing_page_url	https://doi.org/10.3390/math12060815
publication_date	2024-03-10
publication_year	2024
referenced_works	https://openalex.org/W23086234, https://openalex.org/W2037734816, https://openalex.org/W2058540481, https://openalex.org/W1998033277, https://openalex.org/W6607029143, https://openalex.org/W2067819059, https://openalex.org/W2145058185, https://openalex.org/W1981707938, https://openalex.org/W2018193583, https://openalex.org/W2170019535, https://openalex.org/W2130779295, https://openalex.org/W598745901, https://openalex.org/W2159234002, https://openalex.org/W2895079933, https://openalex.org/W6777128930, https://openalex.org/W2055817410, https://openalex.org/W4322745984, https://openalex.org/W3005191414, https://openalex.org/W1981482848, https://openalex.org/W2109339972, https://openalex.org/W2014926578, https://openalex.org/W4295408312, https://openalex.org/W3024688609, https://openalex.org/W171488552
referenced_works_count	24
abstract_inverted_index.a	3
abstract_inverted_index.by	13
abstract_inverted_index.in	6, 29
abstract_inverted_index.is	22
abstract_inverted_index.of	20, 24, 39, 42
abstract_inverted_index.The	17
abstract_inverted_index.and	46, 52
abstract_inverted_index.are	10
abstract_inverted_index.for	55
abstract_inverted_index.has	47
abstract_inverted_index.one	23
abstract_inverted_index.the	25, 36
abstract_inverted_index.This	32
abstract_inverted_index.form	19
abstract_inverted_index.role	5
abstract_inverted_index.Smith	18, 37
abstract_inverted_index.forms	38
abstract_inverted_index.plays	2
abstract_inverted_index.types	41
abstract_inverted_index.which	9
abstract_inverted_index.Matrix	0
abstract_inverted_index.mainly	34
abstract_inverted_index.matrix	56
abstract_inverted_index.topics	28
abstract_inverted_index.article	33
abstract_inverted_index.derived	49
abstract_inverted_index.pivotal	4
abstract_inverted_index.several	40, 50
abstract_inverted_index.matrices	21, 45
abstract_inverted_index.research	27
abstract_inverted_index.systems,	8
abstract_inverted_index.bivariate	43
abstract_inverted_index.important	26
abstract_inverted_index.matrices.	16, 31
abstract_inverted_index.necessary	51
abstract_inverted_index.typically	11
abstract_inverted_index.conditions	54
abstract_inverted_index.polynomial	15, 30, 44
abstract_inverted_index.sufficient	53
abstract_inverted_index.equivalence	1
abstract_inverted_index.represented	12
abstract_inverted_index.equivalence.	57
abstract_inverted_index.investigates	35
abstract_inverted_index.multivariate	14
abstract_inverted_index.successfully	48
abstract_inverted_index.multidimensional	7
cited_by_percentile_year
corresponding_author_ids	https://openalex.org/A5103228907
countries_distinct_count	1
institutions_distinct_count	3
corresponding_institution_ids	https://openalex.org/I121296143
citation_normalized_percentile.value	0.04922326
citation_normalized_percentile.is_in_top_1_percent	False
citation_normalized_percentile.is_in_top_10_percent	False