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books/bookvolbib Axiom Citations in the Literature
Goal: Axiom Literate Programming
\index{Salem, Fatima Khaled Abu}
\begin{chunk}{axiom.bib}
@PhDThesis{Sale04,
author = "Salem, Fatima Khaled Abu",
title = "Factorisation Algorithms for Univariate and Bivariate Polynomials
over Finite Fields",
school = "Meron College",
year = "2004",
paper = "Sale04",
url = "http://www.cs.aub.edu.lb/fa21/Dissertations/My\_thesis.pdf",
abstract =
"In this thesis we address algorithms for polynomial factorisation
over finite fields. In the univariate case, we study a recent
algorithm due to Niederreiter where the factorisation problem is
reduced to solving a linear system over the finite field in question,
and the solutions are used to produce the complete factorisation of
the polynomials into irreducibles. We develop a new algorithm for
solving the linear system using sparse Gaussian elimination with the
Markowitz ordering strategy, and conjecture that the Niederreiter
linear system is not only initially sparse, but also preserves its
sparsity throughout the Gaussian elimination phase. We develop a new
bulk synchronous parallel (BSP) algorithm base on the approach of
Gottfert for extracting the factors of a polynomial using a basis of
the Niederreiter solution set of $\mathbb{F}_2$. We improve upon the
complexity and performance of the original algorithm, and produce
binary univariate factorisations of trinomials up to degree 400000.
We present a new approach to multivariate polynomial factorisation
which incorporates ideas from polyhedral geometry, and generalises
Hensel lifting. The contribution is an algorithm for factoring
bivariate polynomials via polytopes which is able to exploit to some
extent the sparsity of polynomials. We further show that the polytope
method can be made sensitive to the number of nonzero terms of the
input polynomial. We describe a sparse adaptation of the polytope
method over finite fields of prime order which requires fewer bit
operations and memory references for polynomials which are known to be
the product of two sparse factors. Using this method, and to the best
of our knowledge, we achieve a world record in binary bivariate
factorisation of a sparse polynomial of degree 20000. We develop a BSP
variant of the absolute irreducibility testing via polytopes given in
[45], producing a more memory and run time efficient method that can
provide wider ranges of applicability. We achieve absolute
irreducibility testing of a bivariate and trivariate polynomial of
degree 30000, and of multivariate polynomials with up to 3000
variables."
}
\end{chunk}
\index{Gianni, P.}
\index{Trager, B.}
\begin{chunk}{axiom.bib}
@Article{Gian96,
author = "Gianni, P. and Trager, B.",
title = "Square-free algorithms in positive characteristic",
journal =
"J. of Applicable Algebra in Engineering, Communication and Computing",
volume = "7",
pages = "1-14",
year = "1996",
}
\end{chunk}
\index{Shoup, Victor}
\begin{chunk}{axiom.bib}
@InProceedings{Shou91,
author = "Shoup, Victor",
title = "A Fast Deterministic Algorithm for Factoring Polynomials over
Finite Fields of Small Characteristic",
booktitle = "Proc. ISSAC 1991",
series = "ISSAC 1991",
year = "1991",
pages = "14-21",
paper = "Shou91.pdf",
url = "http://www.shoup.net/papers/quadfactor.pdf",
abstract =
"We present a new algorithm for factoring polynomials over finite
fields. Our algorithm is deterministic, and its running time is
``almost'' quadratic when the characteristic is a small fixed
prime. As such, our algorithm is asymptotically faster than previously
known deterministic algorithms for factoring polynomials over finite
fields of small characteristic."
}
\end{chunk}
\index{von zur Gathen, Joachim}
\index{Kaltofen, Erich}
\begin{chunk}{axiom.bib}
@Article{Gath85b,
author = "{von zur Gathen}, Joachim and Kaltofen, E.",
title = "Polynomial-Time Factorization of Multivariate Polynomials over
Finite Fields",
journal = "Math. Comput.",
year = "1985",
volume = "45",
pages = "251-261",
url =
"http://www.math.ncsu.edu/~kaltofen/bibliography/85/GaKa85_mathcomp.ps.gz",
paper = "Gath85.ps",
abstract =
"We present a probabilistic algorithm that finds the irreducible
factors of a bivariate polynomial with coefficients from a finite
field in time polynomial in the input size, i.e. in the degree of the
polynomial and $log$(cardinality of field). The algorithm generalizes
to multivariate polynomials and has polynomial running time for
densely encoded inputs. Also a deterministic version of the algorithm
is discussed whose running time is polynomial in the degree of the
input polynomial and the size of the field."
}
\end{chunk}
\index{von zur Gathen, Joachim}
\index{Panario, Daniel}
\begin{chunk}{axiom.bib}
@Article{Gath01,
author = "von zur Gathen, Joachim and Panario, Daniel",
title = "Factoring Polynomials Over Finite Fields: A Survey",
journal = "J. Symbolic Computation",
year = "2001",
volume = "31",
pages = "3-17",
paper = "Gath01.pdf",
url =
"http://people.csail.mit.edu/dmoshdov/courses/codes/poly-factorization.pdf",
keywords = "survey",
abstract =
"This survey reviews several algorithms for the factorization of
univariate polynomials over finite fields. We emphasize the main ideas
of the methods and provide and up-to-date bibliography of the problem.
This paper gives algorithms for {\sl squarefree factorization},
{\sl distinct-degree factorization}, and {\sl equal-degree factorization}.
The first and second algorithms are deterministic, the third is
probabilistic."
}
\end{chunk}
\index{Augot, Daniel}
\index{Camion, Paul}
\begin{chunk}{axiom.bib}
@Article{Augo97,
author = "Augot, Daniel and Camion, Paul",
title = "On the computation of minimal polynomials, cyclic vectors,
and Frobenius forms",
journal = "Linear Algebra Appl.",
volume = "260",
pages = "61-94",
year = "1997",
keywords = "axiomref",
paper = "Augo97.pdf",
abstract =
"Algorithms related to the computation of the minimal polynomial of an
$x\times n$ matrix over a field $K$ are introduced. The complexity of
the first algorithm, where the complete factorization of the
characteristic polynomial is needed, is $O(\sqrt{n}\cdot n^3)$. An
iterative algorithm for finding the minimal polynomial has complexity
$O(n^3+n^2m^2)$, where $m$ is a parameter of the shift Hessenberg
matrix used. The method does not require the knowlege of the
characteristic polynomial. The average value of $m$ is $O(log n)$.
Next methods are discussed for finding a cyclic vector for a matrix.
The authors first consider the case when its characteristic polynomial
is squarefree. Using the shift Hessenberg form leads to an algorithm
at cost $O(n^3 + n^2m^2)$. A more sophisticated recurrent procedure
gives the result in $O(n^3)$ steps. In particular, a normal basis for
an extended finite field of size $q^n$ will be obtained with complexity
$O(n^3+n^2 log q)$.
Finally, the Frobenius form is obtained with asymptotic average
complexity $O(n^3 log n)$."
}
\end{chunk}
\index{Bernardin, Laurent}
\index{Monagan, Michael B.}
\begin{chunk}{axiom.bib}
@InProceedings{Bern97a,
author = "Bernardin, Laurent and Monagan, Michael B.",
title = "Efficient multivariate factorization over finite fields",
booktitle = "Applied algebra, algebraic algorithms and error-correcting
codes",
series = "AAECC-12",
year = "1997",
location = "Toulouse, France",
publisher = "Springer",
pages = "15-28",
keywords = "axiomref",
paper = "Bern97a.pdf",
url = "http://www.cecm.sfu.ca/~monaganm/papers/AAECC.pdf",
abstract =
"We describe the Maple implementation of multivariate factorization
over general finite fields. Our first implementation is available in
Maple V Release 3. We give selected details of the algorithms and show
several ideas that were used to improve its efficiency. Most of the
improvements presented here are incorporated in Maple V Release 4. In
particular, we show that we needed a general tool for implementing
computations in GF$(p^k)[x_1,x_2,\cdots,x_v]$. We also needed an
efficient implementation of our algorithms $\mathbb{Z}_p[y][x]$ in
because any multivariate factorization may depend on several bivariate
factorizations. The efficiency of our implementation is illustrated by
the ability to factor bivariate polynomials with over a million
monomials over a small prime field."
}
\end{chunk}
\index{Bronstein, Manuel}
\index{Weil, Jacques-Arthur}
\begin{chunk}{axiom.bib}
@Article{Bron97a,
author = "Bronstein, Manuel and Weil, Jacques-Arthur",
title = "On Symmetric Powers of Differential Operators",
series = "ISSAC'97",
year = "1997",
pages = "156-163",
keywords = "axiomref",
url =
"http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html",
paper = "Bro97a.pdf",
publisher = "ACM, NY",
abstract = "
We present alternative algorithms for computing symmetric powers of
linear ordinary differential operators. Our algorithms are applicable
to operators with coefficients in arbitrary integral domains and
become faster than the traditional methods for symmetric powers of
sufficiently large order, or over sufficiently complicated coefficient
domains. The basic ideas are also applicable to other computations
involving cyclic vector techniques, such as exterior powers of
differential or difference operators."
}
\end{chunk}
\index{Calmet, J.}
\index{Campbell, J.A.}
\begin{chunk}{axiom.bib}
@Article{Calm97,
author = "Calmet, J. and Campbell, J.A.",
title = "A perspective on symbolic mathematical computing and
artificial intelligence",
journal = "Ann. Math. Artif. Intell.",
volume = "19",
number = "3-4",
pages = "261-277",
year = "1997",
keywords = "axiomref",
paper = "Calm97.pdf",
url =
"http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.52.5425&rep=rep1&type=pdf",
abstract =
"The nature and history of the research area common to artificial
intelligence and symbolic mathematical computation are examined, with
particular reference to the topics having the greatest current amount
of activity or potential for further development: mathematical
knowledge-based computing environments, autonomous agents and
multi-agent systems, transformation of problem descriptions in logics
into algebraic forms, exploitation of machine learning, qualitative
reasoning, and constraint-based programming. Knowledge representation,
for mathematical knowledge, is identified as a central focus for much
of this work. Several promising topics for further research are stated."
}
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