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Factoring polynomials over global fields I

2005
*
Journal of symbolic computation
*

In this paper we present a generic algorithm for

doi:10.1016/j.jsc.2004.09.006
fatcat:uwllsbftzfbkjbyi5prkxfiepi
*factoring**polynomials**over**global**fields*F. ... Generic*Factorization*Algorithm. Input An integral domain R with quotient*field*K and a square-free*polynomial*g(t) ∈ K [t] of degree greater than one. Step 1. ...*Factorizations*of g(t)*over*R and*over*K can differ substantially. For*global**fields*the concept of algebraic integers can be used to solve this problem. ...##
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Sentences over Integral Domains and Their Computational Complexities

1999
*
Information and Computation
*

As is well known, for all

doi:10.1006/inco.1998.2771
fatcat:yp5tgb7xwzbpdoq4wdocbubjcq
*global**fields*K there also are algorithms for*factoring**polynomials**over*K (Fried and Jarden, 1986) . ...*Factoring**polynomials**over*K with parameters \_ sentences Ring of integers of a*global**field*Co-NP-complete _\ sentences Ring of integers of a*global**field*NP-complete \_ sentences K[T] : ...##
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On invariance of degree for certain computations

2004
*
Journal of Complexity
*

a

doi:10.1016/j.jco.2003.11.007
fatcat:cx5p4ud2yrbpli5saapndapzt4
*polynomial*of degree n: This generalizes results of Cucker, and holds for*polynomials*and machines*over*an infinite ordered topological*field*and in certain cases*over*Z as well. ... We prove that a sequential or parallel machine in the Blum-Shub-Smale model, which recognizes the roots of an irreducible*polynomial*f ðx; yÞ of degree n; in*globally*bounded time T; must actually compute ... Let k be a*field*and f*i*Ak½x 1 ; x 2 ; yx n ;*i*¼ 1; y; m; be*polynomials*in n variables, V ð f Þ ¼ fxAk n j f ðxÞ ¼ 0g and V ðff*i*j 1pipmgÞ ¼ fxAk n j f*i*ðxÞ ¼ 0;*i*¼ 1ymg: Proof. ...##
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Rational separability over a global field

1996
*
Annals of Pure and Applied Logic
*

F such that for every x E RI for some

doi:10.1016/0168-0072(95)00023-2
fatcat:bybgngvf7vczpgumxmi72rn5iq
*i*, Ci(x) E R2. ... elements of R2 do not have*factors*of relative degree 1 in some simple extension of K. ... Lemma 3 . 1 . 31 Let K be a*global**field*and let T(x) = Cik,,aix', where ai E K. Let c = c(T) =*i*(max lordPail),*i*=(J P where p ranges*over*all the non-archimedean primes of K. ...##
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A BSP Parallel Model for the Göttfert Algorithm over F 2
[chapter]

2004
*
Lecture Notes in Computer Science
*

In Niederreiter's

doi:10.1007/978-3-540-24669-5_28
fatcat:vmr44eiowvbndhob65jzpckype
*factorization*algorithm for univariate*polynomials**over*finite*fields*, the*factorization*problem is reduced to solving a linear system*over*the finite*field*in question, and the solutions ... are used to produce the complete*factorization*of the*polynomial*into irreducibles. ... Let f be a*polynomial*of degree d*over*F 2 , and f = g e1 1 ...g em m be its canonical*factorization*of*over*the*field*. Let N f be the Niederreiter matrix of coefficients of f ([1], [2] ). ...##
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Irreducibility of polynomials over global fields is diophantine

2018
*
Compositio Mathematica
*

Given a

doi:10.1112/s0010437x17007977
fatcat:qjiermf22rcivbirjaw7l5a27i
*global**field*$K$ and a positive integer $n$ , we present a diophantine criterion for a*polynomial*in one variable of degree $n$*over*$K$ not to have a root in $K$ . ... We also deduce a diophantine criterion for a*polynomial**over*$K$ of given degree in a given number of variables to be irreducible. ...*I*would like to thank my supervisor, Jochen Koenigsmann, for helpful discussions on the subject of this paper and comments on the exposition. ...##
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Factoring polynomials over global fields II

2005
*
Journal of symbolic computation
*

Also, a generalization of the application of LLL reduction for

doi:10.1016/j.jsc.2005.03.003
fatcat:imklj53lrzecbdm4cuecjldmqi
*factoring**polynomials**over*arbitrary*global**fields*is developed. ... In this paper we describe software for an efficient*factorization*of*polynomials**over**global**fields*F. The algorithm for function*fields*was recently incorporated into our system KANT. ... The*factorization*of*polynomials**over*finite*fields*in Step 2 is not discussed in this paper. ...##
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Computing the Galois group of a polynomial over a p-adic field
[article]

2020
*
arXiv
*
pre-print

We present a family of algorithms for computing the Galois group of a

arXiv:2003.05834v1
fatcat:qw5x7kvepva7bfuno5oiwyx3lq
*polynomial*defined*over*a p-adic*field*. Apart from the "naive" algorithm, these are the first general algorithms for this task. ... Similarly a*global*model for F (x) ∈ K[x] extending*i*is k F k where F = k F k is the*factorization**over*K of F into irreducible*factors*, L k /K are the corresponding extensions,*i*k : L k → L k are*global*...*Factors*This*factorizes*F (x) = k F k (x) into irreducible*factors**over*K, produces a*global*model F k (x) for each*factor*, and then the*global*model is F(x) = k F k (x). ...##
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Finite fields and applications: Proceedings of the third international conference, Glasgow, July 1995

1997
*
Computers and Mathematics with Applications
*

*Factoring*cyclotomic

*polynomials*

*over*large finite

*fields*(Greg Stein). Character sums and coding theory (Serguei A. Stepanov). ... Cellular automata, substitutions and

*factorization*of

*polynomials*

*over*finite

*fields*(Ehler Lange). A new class of two weight codes (Philippe Langevin). ...

##
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Modern compiler implementation in ML: Basic techniques

1997
*
Computers and Mathematics with Applications
*

*Factoring*cyclotomic

*polynomials*

*over*large finite

*fields*(Greg Stein). Character sums and coding theory (Serguei A. Stepanov). ... Cellular automata, substitutions and

*factorization*of

*polynomials*

*over*finite

*fields*(Ehler Lange). A new class of two weight codes (Philippe Langevin). ...

##
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Java language reference

1997
*
Computers and Mathematics with Applications
*

*Factoring*cyclotomic

*polynomials*

*over*large finite

*fields*(Greg Stein). Character sums and coding theory (Serguei A. Stepanov). ... Cellular automata, substitutions and

*factorization*of

*polynomials*

*over*finite

*fields*(Ehler Lange). A new class of two weight codes (Philippe Langevin). ...

##
###
Modern compiler implementation in C: Basic techniques

1997
*
Computers and Mathematics with Applications
*

*Factoring*cyclotomic

*polynomials*

*over*large finite

*fields*(Greg Stein). Character sums and coding theory (Serguei A. Stepanov). ... Cellular automata, substitutions and

*factorization*of

*polynomials*

*over*finite

*fields*(Ehler Lange). A new class of two weight codes (Philippe Langevin). ...

##
###
Modern compiler implementation in Java: Basic techniques

1997
*
Computers and Mathematics with Applications
*

*Factoring*cyclotomic

*polynomials*

*over*large finite

*fields*(Greg Stein). Character sums and coding theory (Serguei A. Stepanov). ... Cellular automata, substitutions and

*factorization*of

*polynomials*

*over*finite

*fields*(Ehler Lange). A new class of two weight codes (Philippe Langevin). ...

##
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Characteristic polynomials of central simple algebras
[article]

2012
*
arXiv
*
pre-print

We characterize characteristic

arXiv:1109.3851v2
fatcat:flbk3ueuundytgw7hygegidv6y
*polynomials*of elements in a central simple algebra. ... We also give an account for the theory of rational canonical forms for separable linear transformations*over*a central division algebra, and a description of separable conjugacy classes of the multiplicative ... When F is a*global**field*, these can be computed explicitly using Brauer groups*over*local*fields*[6] and the period-index relation [ Which*polynomial*is characteristic? ...##
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Polynomial mappings defined by forms with a common factor

1992
*
Séminaire de Théorie des Nombres de Bordeaux
*

If Il is a

doi:10.5802/jtnb.71
fatcat:tsbpw7rxlvdbblqoqmddiu64xm
*global**field*(*i*. e., either an algebraic number*field*or an algebraic function*field*in one variable*over*a finite*field*), then we use the finiteness results from [8, 2.5] . 0 We close this ... Let Kl = Ko(t) be a rational function*field**over*an arbitrary*field*Iio , and let P be the set of all monic irreducible*polynomials*in Proof. ...
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