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| books/bookvolbib Axiom Citations in the Literature | ||
| src/input/Makefile fix typo | ||
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| Goal: Axiom Literate Programming | ||
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| \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. | ||
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| 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." | ||
| } | ||
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| \end{chunk} | ||
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| \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", | ||
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| } | ||
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| \end{chunk} | ||
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| \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." | ||
| } | ||
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| \end{chunk} | ||
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| \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." | ||
| } | ||
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| \end{chunk} | ||
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| \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." | ||
| } | ||
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| \end{chunk} | ||
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| \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 $O(log n)$. | ||
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| 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 | ||
| $O(n^3+n^2 log q)$. | ||
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| Finally, the Frobenius form is obtained with asymptotic average | ||
| complexity $O(n^3 log n)$." | ||
| } | ||
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| \end{chunk} | ||
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| \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." | ||
| } | ||
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| \end{chunk} | ||
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| \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." | ||
| } | ||
|
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| \end{chunk} | ||
|
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| \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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| \end{chunk} | ||
| Goal: Axiom build | ||
|
|
||
| Somewhere along the way I fat-fingered a character delete | ||
| causing the build to break. Sigh. |
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