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Factoring rational polynomials over the complexes

C. Bajaj, J. Canny, R. Garrity, J. Warren
1989 Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation - ISSAC '89  
We give NC algorithms for determining 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.  ... 
doi:10.1145/74540.74551 dblp:conf/issac/BajajCGW89 fatcat:4ipafaazbzc4haplxnpabylbxq

Factoring Rational Polynomials over the Complex Numbers

Chanderjit Bajaj, John Canny, Thomas Garrity, Joe Warren
1993 SIAM journal on computing (Print)  
doi:10.1137/0222024 fatcat:vavblkbdlze23kseio3shtyagm

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

Zhudeng Wang, Qing Wang, Nan Qin
2018 Advances in Linear Algebra and Matrix Theory  
A matrix is similar to Jordan canonical form 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).  ... 
doi:10.4236/alamt.2018.81001 fatcat:vg7c5wkdobbljdb3zigbq65uji

Polynomial factorization

Erich Kaltofen
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] .  ... 
doi:10.1145/860854.860857 dblp:conf/issac/Kaltofen03 fatcat:xeku3tldmbgv7kg3qgeyi3y2r4

Rational Motions with Generic Trajectories of Low Degree [article]

Johannes Siegele, Daniel F. Scharler, Hans-Peter Schröcker
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.  ... 
arXiv:1907.11525v2 fatcat:tddvdhauzrdvbbt4peoyjkjcum

Factoring multivariate polynomials via partial differential equations

Shuhong Gao
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.  ... 
doi:10.1090/s0025-5718-02-01428-x fatcat:btjtelv4nrdhrh5obd6bwb4lb4

A Verified Implementation of Algebraic Numbers in Isabelle/HOL

Sebastiaan J. C. Joosten, René Thiemann, Akihisa Yamada
2018 Journal of automated reasoning  
We moreover provide algorithms that can identify all 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.  ... 
doi:10.1007/s10817-018-09504-w pmid:32226180 pmcid:PMC7089722 fatcat:pvzb5tg36jdb5bfrvnlgmajq7y

Computing the irreducible real factors and components of an algebraicf curve

E. Kaltofen
1989 Proceedings of the fifth annual symposium on Computational geometry - SCG '89  
We show that our algorithms are of 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.  ... 
doi:10.1145/73833.73842 dblp:conf/compgeom/Kaltofen89 fatcat:xjtyaat2gfgfpfwf2recioz4my

Computing the irreducible real factors and components of an algebraic curve

Erich Kaltofen
1990 Applicable Algebra in Engineering, Communication and Computing  
We show that our algorithms are of 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.  ... 
doi:10.1007/bf01810297 fatcat:3p6ivafxrbhzvbmlbz7ftrzwfa

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

Javier Chavarriga, Hector Giacomini, Jaume Giné, Jaume Llibre
2003 Journal of Differential Equations  
We mainly study 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 ?  ... 
doi:10.1016/s0022-0396(03)00190-6 fatcat:w2yiypfdk5apdhyd6tsx4z2rpy

Effective Noether irreducibility forms and applications

Erich Kaltofen
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.  ... 
doi:10.1145/103418.103431 dblp:conf/stoc/Kaltofen91 fatcat:3gg6qt7nobfbta4jzjoyvqqoqm

A note on Gao's algorithm for polynomial factorization

Carlos Hoppen, Virginia M. Rodrigues, Vilmar Trevisan
2011 Theoretical Computer Science  
Shuhong Gao (2003) [6] has proposed an efficient algorithm to 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.  ... 
doi:10.1016/j.tcs.2010.11.048 fatcat:afedgbxfx5d7tpibsxwd3e6xwa

A Jordan factorization theorem for polynomial matrices

H. K. Wimmer
1979 Proceedings of the American Mathematical Society  
It is shown that a 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/?  ... 
doi:10.1090/s0002-9939-1979-0532135-6 fatcat:5nkuztgyf5dzrcphr5rrxpmolq

Effective Noether Irreducibility Forms and Applications

E. Kaltofen
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.  ... 
doi:10.1006/jcss.1995.1023 fatcat:xkaq7zcrujdpthdjwczoionsci
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