Author | Collins, George E. |
Source | ACM Digital Library |
Content type | Text |
Publisher | Association for Computing Machinery (ACM) |
File Format | |
Copyright Year | ©1967 |
Language | English |
Subject Domain (in DDC) | Computer science, information & general works ♦ Data processing & computer science |
Abstract | Let @@@@ be an integral domain, P(@@@@) the integral domain of polynomials over @@@@. Let $\textit{P},$ $\textit{Q}$ ∈ P(@@@@) with $\textit{m}$ @@@@ deg $(\textit{P})$ ≥ $\textit{n}$ = deg $(\textit{Q})$ > 0. Let $\textit{M}$ be the matrix whose determinant defines the resultant of $\textit{P}$ and $\textit{Q}.$ Let $\textit{Mij}$ be the submatrix of $\textit{M}$ obtained by deleting the last $\textit{j}$ rows of $\textit{P}$ coefficients, the last $\textit{j}$ rows of $\textit{Q}$ coefficients and the last $2\textit{j}+1$ columns, excepting column $\textit{m}$ — $\textit{n}$ — $\textit{i}$ — $\textit{j}$ (0 ≤ $\textit{i}$ ≤ $\textit{j}$ < $\textit{n}).$ The polynomial $\textit{Rj}(\textit{x})$ = $∑\textit{i}\textit{i}=0$ det $(\textit{Mij})\textit{xi}$ is the j-t subresultant of $\textit{P}$ and $\textit{Q},$ $\textit{R0}$ being the resultant. If $\textit{b}$ = $£(\textit{Q}),$ the leading coefficient of $\textit{Q},$ then exist uniquely $\textit{R},$ $\textit{S}$ ∈ P(@@@@) such that $\textit{b}\textit{m-n}+1$ $\textit{P}$ = $\textit{QS}$ + $\textit{R}$ with deg $(\textit{R})$ < $\textit{n};$ define $R(\textit{P},$ $\textit{Q})$ = $\textit{R}.$ Define $\textit{Pi}$ ∈ $P(\textit{F}),$ $\textit{F}$ the quotient field of @@@@, inductively: $\textit{P}1$ = $\textit{P},$ $\textit{P}2$ = $\textit{Q},$ $\textit{P}3$ = $R\textit{P}1,$ $\textit{P}2$ $\textit{P}\textit{i}-2$ = $R(\textit{Pi},$ $\textit{P}\textit{i}+1)/\textit{c}δ\textit{i}-1+1\textit{i}$ for $\textit{i}$ ≥ $\textit{2}$ and $\textit{n}\textit{i}+1$ > 0, where $\textit{c}\textit{i}$ = $£(\textit{Pi}),$ $\textit{ni}$ = deg $(\textit{Pi})$ and $δ\textit{i}$ = $\textit{ni}$ — $\textit{n}\textit{i}+1.$ $\textit{P}1,$ $\textit{P}2,$ …, $\textit{Pk},$ for $\textit{k}$ ≥ 3, is called a reduced polynomial remainder sequence. Some of the main results are: (1) $\textit{Pi}$ ∈ P(@@@@) for 1 ≤ $\textit{i}$ ≤ $\textit{k};$ (2) $\textit{Pk}$ = ± $\textit{AkR}\textit{n}\textit{k}-1-1,$ when $\textit{Ak}$ = $&Pgr;\textit{k}-2\textit{i}-2\textit{c}δ\textit{i}-1(δ\textit{i}-1)\textit{i};$ (3) $\textit{c}δ\textit{k}-1-1\textit{k}$ $\textit{Pk}$ = $±\textit{A}\textit{k}+1\textit{R}\textit{n}\textit{k};$ (4) $\textit{Rj}$ = 0 for $\textit{nk}$ < $\textit{j}$ < $\textit{n}\textit{k}-1$ — 1. Taking @@@@ to be the integers $\textit{I},$ or $P\textit{r}(\textit{I}),$ these results provide new algorithms for computing resultant or greatest common divisors of univariate or multivariate polynomials. Theoretical analysis and extensive testing on a high-speed computer show the new g.c.d. algorithm to be faster than known algorithms by a large factor. When applied to bivariate polynomials, for example this factor grows rapidly with the degree and exceeds 100 in practical cases. |
ISSN | 00045411 |
Age Range | 18 to 22 years ♦ above 22 year |
Educational Use | Research |
Education Level | UG and PG |
Learning Resource Type | Article |
Publisher Date | 1967-01-01 |
Publisher Place | New York |
e-ISSN | 1557735X |
Journal | Journal of the ACM (JACM) |
Volume Number | 14 |
Issue Number | 1 |
Page Count | 15 |
Starting Page | 128 |
Ending Page | 142 |
Ministry of Human Resource Development (MHRD) under its National Mission on Education through Information and Communication Technology (NMEICT) has initiated the National Digital Library of India (NDLI) project to develop a framework of virtual repository of learning resources with a single-window search facility. Filtered and federated searching is employed to facilitate focused searching so that learners can find out the right resource with least effort and in minimum time. NDLI is designed to hold content of any language and provides interface support for leading vernacular languages, (currently Hindi, Bengali and several other languages are available). It is designed to provide support for all academic levels including researchers and life-long learners, all disciplines, all popular forms of access devices and differently-abled learners. It is being developed to help students to prepare for entrance and competitive examinations, to enable people to learn and prepare from best practices from all over the world and to facilitate researchers to perform inter-linked exploration from multiple sources. It is being developed at Indian Institute of Technology Kharagpur.
NDLI is a conglomeration of freely available or institutionally contributed or donated or publisher managed contents. Almost all these contents are hosted and accessed from respective sources. The responsibility for authenticity, relevance, completeness, accuracy, reliability and suitability of these contents rests with the respective organization and NDLI has no responsibility or liability for these. Every effort is made to keep the NDLI portal up and running smoothly unless there are some unavoidable technical issues.
Ministry of Human Resource Development (MHRD), through its National Mission on Education through Information and Communication Technology (NMEICT), has sponsored and funded the National Digital Library of India (NDLI) project.
Phone: +91-3222-282435
For any issue or feedback, please write to ndl-support@iitkgp.ac.in