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On the genericity of the modular polynomial GCD algorithm

1999
*
Proceedings of the 1999 international symposium on Symbolic and algebraic computation - ISSAC '99
*

In this paper we study

doi:10.1145/309831.309861
dblp:conf/issac/KaltofenM99
fatcat:b3d3tgpkpneybig62duxhzgp2a
*the**generic*setting*of**the**modular**GCD**algorithm*. We develop*the**algorithm*for multivariate*polynomials*over Euclidean domains which have a special kind*of*remainder function. ... Applying this*generic**algorithm*to a*GCD*problem in Z/(p)[t] [x] where p is small yields an improved asymptotic performance over*the*usual approach, and a very practical*algorithm*for*polynomials*over ... Barry Trager shared*the*Axiom code for*the**modular**GCD**algorithm*with us for comparison. ...##
###
On square-free factorization of multivariate polynomials over a finite field

1997
*
Theoretical Computer Science
*

In this paper we present a new deterministic

doi:10.1016/s0304-3975(97)00059-5
fatcat:3zucsxzip5dchdpeayu5sijx2m
*algorithm*for computing*the*square-free decomposition*of*multivariate*polynomials*with coefficients from a finite field. ...*The*new*algorithm*is also simpler to implement and it can rely*on*any existing*GCD**algorithm*without having to worry about choosing "good" evaluation points. ...*The*resulting*algorithm*will, in*general*, be much more efficient than*algorithms*based*on*Musser's as*the**GCD*computations involve*polynomials**of*smaller degree. ...##
###
High-performance polynomial GCD computations on graphics processors

2011
*
2011 International Conference on High Performance Computing & Simulation
*

Our approach exhibits block structure to distribute

doi:10.1109/hpcsim.2011.5999827
dblp:conf/ieeehpcs/Emeliyanenko11
fatcat:saz6fddhqjbgtdbslwpmvojdyi
*the*computation*of*a single*modular**GCD*over several thread blocks, and thus to remove any hardware limitations*on**the*maximal size*of**polynomials*that ... We propose an*algorithm*to compute a greatest common divisor (*GCD*)*of*univariate*polynomials*with large integer coefficients*on*Graphics Processing Units (GPUs). ... IMPLEMENTATION We begin with a*general*framework*of**the**algorithm*. Next, we briefly discuss*the**modular*arithmetic*on**the*GPU. ...##
###
On Degrees of Modular Common Divisors and the Big PrimegcdAlgorithm

2016
*
International Journal of Mathematics and Mathematical Sciences
*

Our modifications are based

doi:10.1155/2016/3262450
fatcat:emgpqyzbf5amjpy6xvc2eod5tm
*on*bounds*of*degrees*of**modular*common divisors*of**polynomials*,*on*estimates*of**the*number*of*prime divisors*of*a resultant, and*on*finding preliminary bounds*on*degrees*of*... To illustrate*the*ideas we apply*the*constructed*algorithms**on*certain*polynomials*, in particular*on**polynomials*from Knuth's example*of*intermediate expression swell. ... Acknowledgments*The*author was supported in part by joint grant 15RF-054*of*RFBR and SCS MES RA (in frames*of*joint research projects SCS and RFBR) and by 15T-1A258 grant*of*SCS MES RA. ...##
###
A Modular Algorithm for Computing Polynomial GCDs over Number Fields presented with Multiple Extensions
[article]

2016
*
arXiv
*
pre-print

Encarnacion also showed how to use rational number to make

arXiv:1601.01038v1
fatcat:svnzapkrbzb23nhfs7m6gkt62i
*the**algorithm*for Q(alpha) output sensitive, that is,*the*number*of*primes used depends*on**the*size*of**the*integers in*the**gcd*and not*on*bounds ... Our fourth contribution is an implementation*of**the**modular**GCD**algorithm*in Maple and in Magma. ... We also acknowlege Mark Moreno Maza for providing details*of*his implementation*of**the*fraction free*algorithm*in [15] . ...##
###
On degrees of modular common divisors and the Big prime gcd algorithm
[article]

2014
*
arXiv
*
pre-print

We consider a few modifications

arXiv:1407.5898v1
fatcat:2t7blorqmzcxvizienlfktb7t4
*of**the*Big prime*modular**algorithm*for*polynomials*in [x]. ... Our modifications are based*on*bounds*of*degrees*of**modular*common divisors*of**polynomials*,*on*estimates*of**the*number*of*prime divisors*of*a resultant and*on*finding preliminary bounds*on*degrees*of*common ...*The*Big prime*modular**gcd**algorithm*is*one**of**the*first and most popular*algorithms**of*computer algebra. ...##
###
Algebraic algorithms in GF(q)

1985
*
Discrete Mathematics
*

Karlsruhe)

doi:10.1016/0012-365x(85)90017-2
fatcat:cwm6z3l5h5hfxglqxitvdwzq5u
*on*algebraic*algorithms*for computing in large Galois Fields GF(q) with q = p" where p is*the*characteristic*of**the*field and may be arbitrarily large. ... This work is materialized by a module*of**algorithms*implemented in*the*ALDES/SACZ computer algebra system, which will be available with*the*next release*of*this system. ... Acknowledgment*The*author is very grateful for*the*hospitality at*the*University*of*Karlsruhe where this work was completed. ...##
###
The computation of polynomial greatest common divisors over an algebraic number field

1989
*
Journal of symbolic computation
*

A

doi:10.1016/s0747-7171(89)80053-7
fatcat:zvignluswjfb5bqa3efsbiha3q
*modular**algorithm*independently developed by Brown (1971) and Collins (1972) eliminates tile problem*of*coefficient growth in*polynomial**gcd*computation over*the*rational integers Z. ... We present a*modular**algorithm*for computing*the*greatest common divisor*of*two*polynomials*over an algebraic number field. Our*algorithm*is an application*of*ideas*of*Brown and Collins. ... From ~he*modular*representation*of*this subresultant PRS*the*degree*of**the**gcd*over Q(a) is determined, using a partially*modular*method (a*modular**algorithm*for*the**gcd**of*univariate integral*polynomials*...##
###
Binary GCD algorithm for computing error locator polynomials in Reed-Solomon decoding

2005
*
Electronics Letters
*

*The*binary

*GCD*

*algorithm*, discovered by Stein, is an alternative to

*the*Euclidean

*algorithm*for computing

*the*greatest common divisor

*of*two integers. ... In this work,

*the*binary

*GCD*

*algorithm*is applied to Reed-Solomon decoding and a novel iterative

*algorithm*for computing error locator

*polynomials*is proposed. ... Evaluation: Compared to

*the*computation

*of*error locator

*polynomials*based

*on*Euclid's

*algorithm*[1, 2] ,

*the*binary

*GCD*-based

*one*requires a similar number

*of*AND and XOR operations for updating r(x) ...

##
###
Book reports

2005
*
Computers and Mathematics with Applications
*

Book Reports section is a regular feature

doi:10.1016/j.camwa.2005.02.001
fatcat:twqxvijfkrdx7gnr26gmsbuzvu
*of*Computers ~ Mathematics with Applications. ... Available online at www.sciencedirect.com computers & .o,=.o. mathematics with applications Computers and Mathematics with Applications 49 (2005) 953-962 www.elsevier.com/locate/camwa BOOK REPORTS*The*...*The*resultant and*gcd*computation. 6.1 Coefficient growth in*the*Euclidean*Algorithm*. 6.2 Gaul3' lemma. 6.3*The*resultant. 6.4*Modular**gcd**algorithms*. 6.5*Modular**gcd**algorithm*in F[x,y]. 6.6 Mignotte's ...##
###
Probabilistic algorithms for sparse polynomials
[chapter]

1979
*
Lecture Notes in Computer Science
*

*The*probabilistic

*algorithm*presen[ed here will be a variation

*of*

*the*

*modular*

*GCD*algorit, hm. ... In [ii], we present a formulation

*of*Hensel's lemma that is somewhat more

*general*than

*the*

*one*in current use and our probabilistic analogue to it. Here, we shall only present

*the*

*modular*

*algorithm*. ...

*The*

*polynomials*that were fed to

*the*various

*GCD*routines were dill and dig~. 2. 1 . 1 Overview

*of*Sparse

*Modular*

*Algorithm*. ...

##
###
Evaluation of the heuristic polynomial GCD

1995
*
Proceedings of the 1995 international symposium on Symbolic and algebraic computation - ISSAC '95
*

*The*Heuristic

*Polynomial*

*GCD*procedure (GCDHEU) is Richard Zippel, Effective

*Polynomial*Computation, Kluwer Academic Publishers, Boston 1993. ... Acknowledgments We thank Daniel Lichtblau

*of*Wolfram Research Inc. for his information

*on*

*the*

*GCD*methods used in Mathematical. ... We also thank Keith Geddes for his encouragement and explanations

*of*

*the*fine details in Maple. ...

##
###
Polynomial factorization over ${\mathbb F}_2$

2002
*
Mathematics of Computation
*

They allow

doi:10.1090/s0025-5718-02-01421-7
fatcat:qoigim4n6nbjpi2hsdxsuomjpe
*polynomials**of*degree up to 250 000 to be factored in about*one*day*of*CPU time, distributing*the*work*on*two processors. ... We describe*algorithms*for*polynomial*factorization over*the*binary field F 2 , and their implementation. ... However, for q = 2,*modular*squaring is cheaper than a*general**modular*multiplication. • D(n),*the*cost for*one*division with remainder*of*two*polynomials**of*degree at most n. • G(n),*the*cost for*one*...##
###
Toward high-performance polynomial system solvers based on triangular decompositions

2010
*
ACM SIGSAM Bulletin
*

Important observation

doi:10.1145/1823931.1823956
fatcat:xqix6emmrfgndf7rqvsspfywpu
*of*this operation. ◮*Modular*multiplication is efficiency-critical to many other operations (*GCD*, inversion, Hensel Lifting), which are themselves*the*major sub-*algorithm**of**polynomial*... scale. ◮ We have reported new*algorithms*, i.e.*modular*multiplication, regular*GCD*, and regularity test. ◮ In this research, we have focused*on**algorithms*modulo regular chains in dimension-zero. ... PART V / Regular chain and*GCD*◮ Let T ⊂ k[x 1 < · · · < x n ] \ k be a triangular set, hence*the**polynomials**of*T have pairwise distinct main variables. ◮*One*can compute T 1 , . . . , T e and G 1 , . ...##
###
APPROXIMATE GCD OF MULTIVARIATE POLYNOMIALS

2000
*
Computer Mathematics
*

We describe

doi:10.1142/9789812791962_0002
fatcat:3up2ac2i7vgspjz2g2igdwhpx4
*algorithms*for computing*the*greatest common divisor*of*two multivariate*polynomials*with inexactly known coefficients. ... We focus*on*extending standard exact EZ-*GCD**algorithm*to an efficient and stable*algorithm*in approximate case. ... Now*the*coefficients*of**the**GCDs*we computed will be*polynomials*in all variables and can be interpolated*one*by*one*. ...
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