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Implementation of prime decomposition of polynomial ideals over small finite fields

Masayuki Noro, Kazuhiro Yokoyama
2004 Journal of symbolic computation  
An algorithm for the prime decomposition 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.  ... 
doi:10.1016/j.jsc.2003.08.004 fatcat:xskdukgarrdpjeuqz3eksrfhwq

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

L{ászl{ó Babai, Lajos R{ónyai
1990 Mathematics of Computation  
We consider the bit-complexity 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] .  ... 
doi:10.1090/s0025-5718-1990-1035925-1 fatcat:theuji7o7bd77fribcpi34apqe

Computing Irreducible Representations of Finite Groups

Laszlo Babai, Lajos Ronyai
1990 Mathematics of Computation  
We consider the bit-complexity 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] .  ... 
doi:10.2307/2008443 fatcat:fqjtiyjmu5cgjkkwdbt3ha7pom

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  ... 
doi:10.1016/s0747-7171(04)00109-9 fatcat:q3cckydpknhjhinygacsvlj52y

Standard Lattices of Compatibly Embedded Finite Fields

Luca De Feo, Hugues Randriam, Édouard Rousseau
2019 Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation - ISSAC '19  
Lattices 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.  ... 
doi:10.1145/3326229.3326251 dblp:conf/issac/FeoRR19 fatcat:apiarmycvbgmrkubhhrhkzc5ie

Computation of unirational fields

Jaime Gutierrez, David Sevilla
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.  ... 
doi:10.1016/j.jsc.2005.05.009 fatcat:jzzzc2rm6jajppgklfkxuzry2m

Computation of unirational fields (extended abstract) [article]

Jaime Gutierrez, David Sevilla
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  ... 
arXiv:0804.1707v1 fatcat:4otz2bh2wzapdmux4hpp6vh64q

Computing the structure of finite algebras

Lajos Rónyai
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.  ... 
doi:10.1016/s0747-7171(08)80017-x fatcat:wubogoa7ljbzpj6hatd5df24je

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

Yao Sun, Dingkang Wang
2010 arXiv   pre-print
A new 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.  ... 
arXiv:0907.2300v2 fatcat:2wmvxidei5bd3mkldrvu3jotv4

Computing primitive elements of extension fields

Kazuhiro Yokoyama, Masayuki Noro, Taku Takeshima
1989 Journal of symbolic computation  
Several mathematical results and new 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.  ... 
doi:10.1016/s0747-7171(89)80061-6 fatcat:qqjvd5nylzbz3h3vkr72iiirse

Algebraic independence over positive characteristic: New criterion and applications to locally low-algebraic-rank circuits

Anurag Pandey, Nitin Saxena, Amit Sinhababu
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.  ... 
doi:10.1007/s00037-018-0167-5 fatcat:7ki7km76ynbulnnbmjavvebuku

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

Yao Sun, DingKang Wang
2013 Science China Mathematics  
An 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  ... 
doi:10.1007/s11425-013-4586-0 fatcat:x6oc7zibxfbovnx7a4xiuuk74m

Efficient Decomposition of Associative Algebras over Finite Fields

W. Eberly, M. Giesbrecht
2000 Journal of symbolic computation  
We also show how to 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.  ... 
doi:10.1006/jsco.1999.0308 fatcat:myr6prcksrge3fufgqbxa6rlsa
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