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Factoring polynomials over global fields I
2005
Journal of symbolic computation
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In this paper we present a generic algorithm for 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. ...
doi:10.1016/j.jsc.2004.09.006
fatcat:uwllsbftzfbkjbyi5prkxfiepi
Sentences over Integral Domains and Their Computational Complexities
1999
Information and Computation
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As is well known, for all 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] : ...
doi:10.1006/inco.1998.2771
fatcat:yp5tgb7xwzbpdoq4wdocbubjcq
On invariance of degree for certain computations
2004
Journal of Complexity
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a 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. ...
doi:10.1016/j.jco.2003.11.007
fatcat:cx5p4ud2yrbpli5saapndapzt4
Rational separability over a global field
1996
Annals of Pure and Applied Logic
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F such that for every x E RI for some 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. ...
doi:10.1016/0168-0072(95)00023-2
fatcat:bybgngvf7vczpgumxmi72rn5iq
A BSP Parallel Model for the Göttfert Algorithm over F 2
[chapter]
2004
Lecture Notes in Computer Science
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In Niederreiter's 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] ). ...
doi:10.1007/978-3-540-24669-5_28
fatcat:vmr44eiowvbndhob65jzpckype
Irreducibility of polynomials over global fields is diophantine
2018
Compositio Mathematica
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2016 pre-print arXiv:1601.07829v2 fatcat:nz2pkrdg6bgmrinejetzwy5nnmOther Versions
Given a 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. ...
doi:10.1112/s0010437x17007977
fatcat:qjiermf22rcivbirjaw7l5a27i
Factoring polynomials over global fields II
2005
Journal of symbolic computation
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Also, a generalization of the application of LLL reduction for 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. ...
doi:10.1016/j.jsc.2005.03.003
fatcat:imklj53lrzecbdm4cuecjldmqi
Computing the Galois group of a polynomial over a p-adic field
[article]
2020
arXiv
pre-print
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We present a family of algorithms for computing the Galois group of a 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). ...
arXiv:2003.05834v1
fatcat:qw5x7kvepva7bfuno5oiwyx3lq
Finite fields and applications: Proceedings of the third international conference, Glasgow, July 1995
1997
Computers and Mathematics with Applications
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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). ...
doi:10.1016/s0898-1221(97)84608-1
fatcat:qpixyynpendb7fhxzgsafywds4
Modern compiler implementation in ML: Basic techniques
1997
Computers and Mathematics with Applications
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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). ...
doi:10.1016/s0898-1221(97)84606-8
fatcat:qlv6mknf5fcdpa3nt5op6xbdae
Java language reference
1997
Computers and Mathematics with Applications
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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). ...
doi:10.1016/s0898-1221(97)84607-x
fatcat:geziy2oxr5bfdioseeieaxsq7m
Modern compiler implementation in C: Basic techniques
1997
Computers and Mathematics with Applications
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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). ...
doi:10.1016/s0898-1221(97)84605-6
fatcat:gzexbpqubzgoxmtaialyqipj3a
Modern compiler implementation in Java: Basic techniques
1997
Computers and Mathematics with Applications
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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). ...
doi:10.1016/s0898-1221(97)84609-3
fatcat:bdcnjqkjizgpdcsw6tkrpkvku4
Characteristic polynomials of central simple algebras
[article]
2012
arXiv
pre-print
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We characterize characteristic 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? ...
arXiv:1109.3851v2
fatcat:flbk3ueuundytgw7hygegidv6y
Polynomial mappings defined by forms with a common factor
1992
Séminaire de Théorie des Nombres de Bordeaux
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If Il is a 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. ...
doi:10.5802/jtnb.71
fatcat:tsbpw7rxlvdbblqoqmddiu64xm
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