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Implementation of prime decomposition of polynomial ideals over small finite fields
2004
Journal of symbolic computation
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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
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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
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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
1990
Mathematics of Computation
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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
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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
2019
Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation - ISSAC '19
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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
2006
Journal of symbolic computation
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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]
2008
arXiv
pre-print
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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
1990
Journal of symbolic computation
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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]
2010
arXiv
pre-print
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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
1989
Journal of symbolic computation
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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
2018
Computational Complexity
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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
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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
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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
2000
Journal of symbolic computation
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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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