An extended GCRD algorithm for parametric univariate polynomial matrices and application to parametric Smith form

doi:10.1016/j.jsc.2022.07.006

	An extended GCRD algorithm for parametric univariate polynomial matrices and application to parametric Smith form
	Wang, Dingkang 1,2; Wang, Hesong 1,2; Wei, Jingjing 1,2; Xiao, Fanghui 3
刊名	JOURNAL OF SYMBOLIC COMPUTATION
	2023-03-01
卷号	115 页码:248-265
关键词	Extended greatest common right divisor Parametric univariate polynomial matrix Comprehensive Gr?bner system Smith normal form
ISSN号	0747-7171
DOI	10.1016/j.jsc.2022.07.006
英文摘要	The first extended greatest common right divisor (GCRD) algorithm for parametric univariate polynomial matrices is presented. The starting point of this GCRD algorithm is the free property of submodules over univariate polynomial rings. We convert the computation of GCRDs to that of free basis for modules and prove that a free basis of the submodule generated by row vectors of input matrices forms just a GCRD of these matrices. The GCRD algorithm is obtained by computing a minimal Grobner basis for the corresponding submodule since a minimal Grobner basis of submodules is a free basis for univariate cases. While the key idea of extended algorithm is to construct a special module by adding the unit vectors which can record the representation coefficients. This method based on modules can be naturally generalized to the parametric case because of the comprehensive Grobner systems for modules. As a consequence, we obtain an extended GCRD algorithm for parametric univariate polynomial matrices. More importantly, we apply the proposed extended GCD algorithm for univariate polynomials (as a special case of matrices) to the computation of Smith normal form, and give the first algorithm for reducing a univariate polynomial matrix with parameters to its Smith normal form. (C) 2022 Elsevier Ltd. All rights reserved.
资助项目	National Natural Science Foundation of China[12171469] ; CAS Key Project[QYZDJ-SSW-SYS022] ; National Key Research and Development Project[2020YFA0712300]
WOS研究方向	Computer Science ; Mathematics
语种	英语
出版者	ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD
WOS记录号	WOS:000860188200011
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/60905]
专题	中国科学院数学与系统科学研究院
通讯作者	Wang, Dingkang
作者单位	1.Chinese Acad Sci, Acad Math & Syst Sci, KLMM, Beijing 100190, Peoples R China 2.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China 3.Hunan Normal Univ, MOE LCSM, Sch Math & Stat, Hunan 410081, Peoples R China
推荐引用方式 GB/T 7714	Wang, Dingkang,Wang, Hesong,Wei, Jingjing,et al. An extended GCRD algorithm for parametric univariate polynomial matrices and application to parametric Smith form[J]. JOURNAL OF SYMBOLIC COMPUTATION,2023,115:248-265.
APA	Wang, Dingkang,Wang, Hesong,Wei, Jingjing,&Xiao, Fanghui.(2023).An extended GCRD algorithm for parametric univariate polynomial matrices and application to parametric Smith form.JOURNAL OF SYMBOLIC COMPUTATION,115,248-265.
MLA	Wang, Dingkang,et al."An extended GCRD algorithm for parametric univariate polynomial matrices and application to parametric Smith form".JOURNAL OF SYMBOLIC COMPUTATION 115(2023):248-265.