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Factoring rational polynomials over the complexes

1989
*
Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation - ISSAC '89
*

We give NC algorithms for determining

doi:10.1145/74540.74551
dblp:conf/issac/BajajCGW89
fatcat:4ipafaazbzc4haplxnpabylbxq
*the*number and degrees of*the*absolute*factors*(*factors*irreducible*over**the**complex*numbers C) of a multivariate*polynomial*with*rational*coefficients. ... These methods rely on*the*fact that*the*connected components of a*complex*hypersurface P(zl, . . . , zn) = 0 minus its singular points correspond to*the*absolute*factors*of P. ... Methods for*factoring**polynomials*with*rational*coefficients*over**the**rational*numbers are wellknown. ...##
###
Factoring Rational Polynomials over the Complex Numbers

1993
*
SIAM journal on computing (Print)
*

##
###
Quasi-Rational Canonical Forms of a Matrix over a Number Field

2018
*
Advances in Linear Algebra and Matrix Theory
*

A matrix is similar to Jordan canonical form

doi:10.4236/alamt.2018.81001
fatcat:vg7c5wkdobbljdb3zigbq65uji
*over**the**complex*field and*the**rational*canonical form*over*a number field, respectively. ... In this paper, we further study*the**rational*canonical form of a matrix*over*any number field. We firstly discuss*the*elementary divisors of a matrix*over*a number field. ... Acknowledgements This work is funded by*the*Flagship Major Development of Jiangsu Higher Education Institution (PPZY2015C211) and College Students Practice Innovation Training Program (201610324027Y). ...##
###
Polynomial factorization

2003
*
Proceedings of the 2003 international symposium on Symbolic and algebraic computation - ISSAC '03
*

*The*problem of

*factoring*a

*polynomial*in a single or several variables

*over*a finite field,

*the*

*rational*numbers or

*the*

*complex*numbers is one of

*the*success stories in

*the*discipline of symbolic computation ...

*Polynomial*-time

*complexity*for

*rational*coefficients was established in

*the*early 1980s by

*the*now-famous lattice basis reduction algorithm of A. Lenstra, H. W. Lenstra, Jr., and L. Lovász. ... [20, 21] , and

*the*deterministic distinct degree

*factorization*for multivariate

*polynomials*

*over*large finite fields [7] . ...

##
###
Rational Motions with Generic Trajectories of Low Degree
[article]

2019
*
arXiv
*
pre-print

*The*trajectories of a

*rational*motion given by a

*polynomial*of degree n in

*the*dual quaternion model of rigid body displacements are generically of degree 2n. ... Our characterizations allow

*the*systematic construction of

*rational*motions with exceptional degree reduction and explain why

*the*trajectory degrees of a

*rational*motion and its inverse motion can be different ...

*Polynomials*

*over*rings come with

*the*notions of left/right evaluation, left/right zeros, and left/right

*factors*. ...

##
###
Factoring multivariate polynomials via partial differential equations

2002
*
Mathematics of Computation
*

*the*

*polynomial*to be

*factored*, and any basis for

*the*solution space gives a complete

*factorization*by computing gcd's and by

*factoring*univariate

*polynomials*

*over*

*the*ground field. ...

*The*new method finds absolute and

*rational*

*factorizations*simultaneously and is easy to implement for finite fields, local fields, number fields, and

*the*

*complex*number field. ...

*The*author thanks Janez Ales, Jürgen Gerhard, Erich Kaltofen, Michael Monagan and Virgínia Rodrigues for their helpful comments and discussions about

*the*paper. ...

##
###
A Verified Implementation of Algebraic Numbers in Isabelle/HOL

2018
*
Journal of automated reasoning
*

We moreover provide algorithms that can identify all

doi:10.1007/s10817-018-09504-w
pmid:32226180
pmcid:PMC7089722
fatcat:pvzb5tg36jdb5bfrvnlgmajq7y
*the*real or*complex*roots of*rational**polynomials*, and two implementations to display algebraic numbers, an approximative version and an injective precise ...*The*development combines various existing formalizations such as matrices, Sturm's theorem, and*polynomial**factorization*, and it includes new formalizations about bivariate*polynomials*, unique*factorization*... , and reproduction in any medium, provided you give appropriate credit to*the*original author(s) and*the*source, provide a link to*the*Creative Commons license, and indicate if changes were made. ...##
###
Computing the irreducible real factors and components of an algebraicf curve

1989
*
Proceedings of the fifth annual symposium on Computational geometry - SCG '89
*

We show that our algorithms are of

doi:10.1145/73833.73842
dblp:conf/compgeom/Kaltofen89
fatcat:xjtyaat2gfgfpfwf2recioz4my
*polynomial*bit*complexity*in*the*degree of*the*equation and*the*size of its coefficients. ... Our construction is based on computing*the*irreducible*complex**factors*and then investigating high precision*complex*floating point coefficients of these*factors*and*the**complex*norms. ... Therefore, such algorithms do not solve our problem in*polynomial*-time. Our*polynomial*-time solution relies on*the*theory of*factoring**polynomials**over**the**complex*numbers. ...##
###
Computing the irreducible real factors and components of an algebraic curve

1990
*
Applicable Algebra in Engineering, Communication and Computing
*

We show that our algorithms are of

doi:10.1007/bf01810297
fatcat:3p6ivafxrbhzvbmlbz7ftrzwfa
*polynomial*bit*complexity*in*the*degree of*the*equation and*the*size of its coefficients. ... Our construction is based on computing*the*irreducible*complex**factors*and then investigating high precision*complex*floating point coefficients of these*factors*and*the**complex*norms. ... Therefore, such algorithms do not solve our problem in*polynomial*-time. Our*polynomial*-time solution relies on*the*theory of*factoring**polynomials**over**the**complex*numbers. ...##
###
Page 490 of Mathematical Reviews Vol. , Issue 98A
[page]

1998
*
Mathematical Reviews
*

Let k(X) be

*the**rational*functions*over*some field k and let & be*the*k-subalgebra of proper*rational*func- tions. ...*The*main idea of*the*authors is to use*the*partial fraction decomposition R ~ , #,, where g runs*over**the*monic irre- ducible*polynomials*and #, denotes*the*k-subalgebra of*rational*functions f/g' € #. ...##
###
Darboux integrability and the inverse integrating factor

2003
*
Journal of Differential Equations
*

We mainly study

doi:10.1016/s0022-0396(03)00190-6
fatcat:w2yiypfdk5apdhyd6tsx4z2rpy
*polynomial*differential systems of*the*form dx=dt ¼ Pðx; yÞ; dy=dt ¼ Qðx; yÞ; where P and Q are*complex**polynomials*in*the*dependent*complex*variables x and y; and*the*independent variable ... We assume that*the**polynomials*P and Q are relatively prime and that*the*differential system has a Darboux first integral of*the*form ...*factorize**the**polynomials*f i which appear in (3) as product of irreducible*factors**over*C½x; y; and take into account that if h and g are*polynomials*and g r 1 1 ? ...##
###
Effective Noether irreducibility forms and applications

1991
*
Proceedings of the twenty-third annual ACM symposium on Theory of computing - STOC '91
*

*the*

*complex*

*factors*of a multivariate integral

*polynomial*, and how to count

*the*number of absolutely irreducible

*factors*of a multivariate

*polynomial*with coefficients in a

*rational*function field, both ... A (multivariate)

*polynomial*

*over*a specific field is said to be absolutely irreducible if it is irreducible

*over*

*the*algebraic closure of its coefficient field. ...

*Factoring*

*over*

*the*

*Complex*Numbers We now present several

*complexity*results for

*factoring*multivariate

*polynomials*

*over*certain algebraically closed fields. ...

##
###
A note on Gao's algorithm for polynomial factorization

2011
*
Theoretical Computer Science
*

Shuhong Gao (2003) [6] has proposed an efficient algorithm to

doi:10.1016/j.tcs.2010.11.048
fatcat:afedgbxfx5d7tpibsxwd3e6xwa
*factor*a bivariate*polynomial*f*over*a field F. ... Moreover, we identify a second vector subspace of G that leads to an analogous theory for*the**rational**factorization*of f . (C. Hoppen), virginia@pucrs.br (V.M. Rodrigues), trevisan@mat.ufrgs.br (V. ... Acknowledgements*The*authors are indebted to anonymous referees for their valuable comments and suggestions. ...##
###
A Jordan factorization theorem for polynomial matrices

1979
*
Proceedings of the American Mathematical Society
*

It is shown that a

doi:10.1090/s0002-9939-1979-0532135-6
fatcat:5nkuztgyf5dzrcphr5rrxpmolq
*complex**polynomial*matrix M(\) which has a proper*rational*inverse can be*factored*into M(\) = C(A)(A/ -J)BQi) where J is a matrix in Jordan normal form and*the*columns of C(X) consist ... For a proper*rational*matrix W with*factorizations*W(X) -CQJ -J)-}B -M(k)~lP(\) -QQQNQC)-1 it will be proved that C consists of Jordan chains of M and B of Jordan chains of TV. ...*The*entries of vectors in C(X) and of matrices in CX*(A) are*complex**rational*functions. A*complex*function / is called proper*rational*, if it is*the*quotient of two*polynomials*, / = p/q, and deg/? ...##
###
Effective Noether Irreducibility Forms and Applications

1995
*
Journal of computer and system sciences (Print)
*

*the*

*complex*

*factors*of a multivariate integral

*polynomial*, and how to count

*the*number of absolutely irreducible

*factors*of a multivariate

*polynomial*with coefficients in a

*rational*function field, both ... A (multivariate)

*polynomial*

*over*a specific field is said to be absolutely irreducible if it is irreducible

*over*

*the*algebraic closure of its coefficient field. ...

*Factoring*

*over*

*the*

*Complex*Numbers We now present several

*complexity*results for

*factoring*multivariate

*polynomials*

*over*certain algebraically closed fields. ...

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