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Implementation of prime decomposition of polynomial ideals over small finite fields

2004
*
Journal of symbolic computation
*

An algorithm for the prime decomposition

doi:10.1016/j.jsc.2003.08.004
fatcat:xskdukgarrdpjeuqz3eksrfhwq
*of**polynomial*ideals over small*finite**fields*is proposed and implemented on the basis*of*previous work*of*the second author. ... To achieve better performance, several improvements are added to the existing algorithm, with strategies for*computational*flow proposed, based on experimental results. ...*polynomials*over any*algebraic**extension**field**of*L can be calculated. ...##
###
Page 12 of Mathematical Reviews Vol. , Issue 2002B
[page]

2002
*
Mathematical Reviews
*

869 12

*FIELD*THEORY AND*POLYNOMIALS*2002b:12004 12E20 68w30 Shoup, Victor (CH-IBM; Riischlikon)*Efficient**computation**of**minimal**polynomials**in**algebraic**extensions**of**finite**fields*. ... Summary: “New algorithms are presented for*computing*the min- imal*polynomial*over a*finite**field*K*of*a given element*in*an*algebraic**extension**of*K*of*the form K[a] or K[a}[f]. ...##
###
Computing irreducible representations of finite groups

1990
*
Mathematics of Computation
*

We consider the bit-complexity

doi:10.1090/s0025-5718-1990-1035925-1
fatcat:theuji7o7bd77fribcpi34apqe
*of*the problem stated*in*the title. Exact*computations**in**algebraic*number*fields*are performed symbolically. ... We also consider the problem*of*decomposing a given representation 'V*of*the*finite*group G over an*algebraic*number*field*F into absolutely irreducible constituents. ... Semisimple*algebras*over*finite**fields*can be decomposed into the sum*of**minimal*left ideals*in*Las Vegas*polynomial*time [20, 27] . ...##
###
Computing Irreducible Representations of Finite Groups

1990
*
Mathematics of Computation
*

We consider the bit-complexity

doi:10.2307/2008443
fatcat:fqjtiyjmu5cgjkkwdbt3ha7pom
*of*the problem stated*in*the title. Exact*computations**in**algebraic*number*fields*are performed symbolically. ... We also consider the problem*of*decomposing a given representation 'V*of*the*finite*group G over an*algebraic*number*field*F into absolutely irreducible constituents. ... Semisimple*algebras*over*finite**fields*can be decomposed into the sum*of**minimal*left ideals*in*Las Vegas*polynomial*time [20, 27] . ...##
###
Index to Volumes 37 and 38

2004
*
Journal of symbolic computation
*

*of*monomial

*algebras*, 537

*Computing*maximal subgroups

*of*

*finite*groups, 589

*Computing*

*minimal*generators

*of*the ideal

*of*a general projective curve, 295 Corner edge cutting and Dixon A-resultant quotients ... , R.S. and MILLER, J.C., Symbolic

*computation*

*of*exact solutions expressible

*in*hyperbolic and elliptic functions for nonlinear PDEs, 669 BALLICO, E. and ORECCHIA, F.,

*Computing*

*minimal*generators

*of*the ... over small

*finite*

*fields*, 1227 JOSWIG, M. and ZIEGLER, G.M., Convex hulls, oracles, and homology, 1247 KIRSCHENHOFER, P. and THUSWALDNER, J.M., Elements

*of*small norm

*in*Shanks' cubic

*extensions*

*of*imaginary ...

##
###
Standard Lattices of Compatibly Embedded Finite Fields

2019
*
Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation - ISSAC '19
*

Lattices

doi:10.1145/3326229.3326251
dblp:conf/issac/FeoRR19
fatcat:apiarmycvbgmrkubhhrhkzc5ie
*of*compatibly embedded*finite**fields*are useful*in**computer**algebra*systems for managing many*extensions**of*a*finite**field*F p at once. ... Compared to Conway*polynomials*, our construction defines a much larger set*of**field**extensions*from a small pre-*computed*table; however it is provably as inefficient as Conway*polynomials*if one wants ... As expected,*computing*the embeddings takes negligible time*in*comparison to the decoration*of*the*finite**fields*. ...##
###
Computation of unirational fields

2006
*
Journal of symbolic computation
*

*In*this paper we present an algorithm for

*computing*all

*algebraic*intermediate subfields

*in*a separably generated unirational

*field*

*extension*(which

*in*particular includes the zero characteristic case) ... One

*of*the main tools is Gröbner bases theory, see [BW93] . Our algorithm also requires

*computing*

*computing*primitive elements and factoring over

*algebraic*

*extensions*. ... It turns out that by factoring the

*minimal*

*polynomial*

*of*α over E[α], we can

*compute*the intermediate

*fields*

*of*the

*extension*E[α]/E. ...

##
###
Computation of unirational fields (extended abstract)
[article]

2008
*
arXiv
*
pre-print

*In*this paper we present an algorithm for

*computing*all

*algebraic*intermediate subfields

*in*a separably generated unirational

*field*

*extension*(which

*in*particular includes the zero characteristic case) ... One

*of*the main tools is Groebner bases theory. Our algorithm also requires

*computing*

*computing*primitive elements and factoring over

*algebraic*

*extensions*. ... It turns out that by factoring the

*minimal*

*polynomial*

*of*α over E[α], we can

*compute*the intermediate

*fields*

*of*the

*extension*E[α]/E ...

##
###
Computing the structure of finite algebras

1990
*
Journal of symbolic computation
*

*In*this paper we address some algorithmic problems related to

*computations*

*in*finitedimensional associative

*algebras*over

*finite*

*fields*. ... Next, we study the problem

*of*finding zero divisors

*in*

*finite*

*algebras*. We show that this problem is

*in*the same complexity class as the problem

*of*factoring

*polynomials*over

*finite*

*fields*. ... (a) If the

*field*K is a

*finite*

*algebraic*

*extension*

*of*the

*field*F then K is a finitedimensional simple and commutative

*algebra*over F. ...

##
###
An Efficient Algorithm for Factoring Polynomials over Algebraic Extension Field
[article]

2010
*
arXiv
*
pre-print

A new

arXiv:0907.2300v2
fatcat:2wmvxidei5bd3mkldrvu3jotv4
*efficient*algorithm is proposed for factoring*polynomials*over an*algebraic**extension**field*. The*extension**field*is defined by a*polynomial*ring modulo a maximal ideal. ... From these factors, the factorization*of*the*polynomial*over the*extension**field*is obtained. ... However, all the existing algorithms for factoring*polynomials*over*algebraic**extension**field*are not so*efficient*. ...##
###
Computing primitive elements of extension fields

1989
*
Journal of symbolic computation
*

Several mathematical results and new

doi:10.1016/s0747-7171(89)80061-6
fatcat:qqjvd5nylzbz3h3vkr72iiirse
*computational*methods are presented for primitive elements and their*minimal**polynomials**of**algebraic**extension*fidds. ... Finally, for a given*polynomial*f over Q, a new method is presented for*computing*a primitive dement*of*the splitting*field**of*f and its*minimal**polynomial*over Q. 0747-7171/891120553+28 $03.00]0 (~ 1989 ... This is part*of*the work*in*the major R&D*of*the Fifth Generation*Computer*Project, conducted under program set up by MITI. ...##
###
Algebraic independence over positive characteristic: New criterion and applications to locally low-algebraic-rank circuits

2018
*
Computational Complexity
*

*In*a set

*of*linearly dependent

*polynomials*, any

*polynomial*can be written as a linear combination

*of*the

*polynomials*forming a basis. ... This is a modest step towards the open question

*of*fast independence testing, over

*finite*

*fields*, posed

*in*(Dvir, Gabizon, Wigderson FOCS'07). ... We have this notion

*of*separability

*in*case

*of*

*field*

*extensions*as well. An

*algebraic*

*extension*E/F is said to be separable if every element α ∈ E has a

*minimal*

*polynomial*over F that is separable. ...

##
###
Page 2057 of Mathematical Reviews Vol. , Issue 96d
[page]

1996
*
Mathematical Reviews
*

It is shown how a

*minimal**polynomial**of*a solution can be*computed*directly for a known*finite*differential Galois group. ...*In*the*algebraic*case, i.e. when pi and d; are*polynomials*, we present a refined algorithm with higher*efficiency*. ...##
###
An efficient algorithm for factoring polynomials over algebraic extension field

2013
*
Science China Mathematics
*

An

doi:10.1007/s11425-013-4586-0
fatcat:x6oc7zibxfbovnx7a4xiuuk74m
*efficient*algorithm is proposed for factoring*polynomials*over an*algebraic**extension**field*defined by a*polynomial*ring modulo a maximal ideal. ... From its factors, the factorization*of*the*polynomial*over the*extension**field*is obtained. ... Noro and Yokoyama presented algorithms for*computing*prime decomposition*of*radical ideals and factorization*of**polynomials*over*algebraic**extension**field**in*[Noro and Yokoyama, 1996 ,Noro and Yokoyama ...##
###
Efficient Decomposition of Associative Algebras over Finite Fields

2000
*
Journal of symbolic computation
*

We also show how to

doi:10.1006/jsco.1999.0308
fatcat:myr6prcksrge3fufgqbxa6rlsa
*compute*a complete set*of*orthogonal primitive idempotents*in*any associative*algebra*over a*finite**field**in*this same time. ... We present new,*efficient*algorithms for some fundamental*computations*with finitedimensional (but not necessarily commutative) associative*algebras*over*finite**fields*. ... Acknowledgements The authors were supported*in*part by the Natural Sciences and Engineering Research Council*of*Canada. ...
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