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1. A Modular Algorithm to Compute the Resultant of Multivariate Polynomials over Algebraic Number Fields Presented with Multiple Extensions

Author

Ansari, Mahsa, Monagan, Michael, Goos, Gerhard, Series Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Boulier, François, editor, Mou, Chenqi, editor, Sadykov, Timur M., editor, and Vorozhtsov, Evgenii V., editor

Published
2024
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2. Computing GCDs of Multivariate Polynomials over Algebraic Number Fields Presented with Multiple Extensions

Author

Ansari, Mahsa, Monagan, Michael, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Boulier, François, editor, England, Matthew, editor, Kotsireas, Ilias, editor, Sadykov, Timur M., editor, and Vorozhtsov, Evgenii V., editor

Published
2023
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3. Solving Parametric Linear Systems Using Sparse Rational Function Interpolation

Author

Jinadu, Ayoola, Monagan, Michael, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Boulier, François, editor, England, Matthew, editor, Kotsireas, Ilias, editor, Sadykov, Timur M., editor, and Vorozhtsov, Evgenii V., editor

Published
2023
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4. High performance SIMD modular arithmetic for polynomial evaluation

Author

Fortin, Pierre, Fleury, Ambroise, Lemaire, François, and Monagan, Michael

Subjects
Computer Science - Distributed, Parallel, and Cluster Computing, Computer Science - Symbolic Computation
Abstract

Two essential problems in Computer Algebra, namely polynomial factorization and polynomial greatest common divisor computation, can be efficiently solved thanks to multiple polynomial evaluations in two variables using modular arithmetic. In this article, we focus on the efficient computation of such polynomial evaluations on one single CPU core. We first show how to leverage SIMD computing for modular arithmetic on AVX2 and AVX-512 units, using both intrinsics and OpenMP compiler directives. Then we manage to increase the operational intensity and to exploit instruction-level parallelism in order to increase the compute efficiency of these polynomial evaluations. All this results in the end to performance gains up to about 5x on AVX2 and 10x on AVX-512.

Published
2020

7. An Interpolation Algorithm for Computing Dixon Resultants

Author

Jinadu, Ayoola, Monagan, Michael, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Boulier, François, editor, England, Matthew, editor, Sadykov, Timur M., editor, and Vorozhtsov, Evgenii V., editor

Published
2022
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8. Resolving zero-divisors using Hensel lifting

Author

Kluesner, John and Monagan, Michael

Subjects
Computer Science - Symbolic Computation
Abstract

Algorithms which compute modulo triangular sets must respect the presence of zero-divisors. We present Hensel lifting as a tool for dealing with them. We give an application: a modular algorithm for computing GCDs of univariate polynomials with coefficients modulo a radical triangular set over the rationals. Our modular algorithm naturally generalizes previous work from algebraic number theory. We have implemented our algorithm using Maple's RECDEN package. We compare our implementation with the procedure RegularGcd in the RegularChains package., Comment: Shorter version to appear in Proceedings of SYNASC 2017

Published
2018

10. Polynomial Factorization in Maple 2019

Author

Monagan, Michael, Tuncer, Baris, Barbosa, Simone Diniz Junqueira, Editorial Board Member, Filipe, Joaquim, Editorial Board Member, Ghosh, Ashish, Editorial Board Member, Kotenko, Igor, Editorial Board Member, Zhou, Lizhu, Editorial Board Member, Gerhard, Jürgen, editor, and Kotsireas, Ilias, editor

Published
2020
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12. The Complexity and Parallel Implementation of Two Sparse Multivariate Hensel Lifting Algorithms for Polynomial Factorization

Author

Chen, Tian, Monagan, Michael, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Woeginger, Gerhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Boulier, François, editor, England, Matthew, editor, Sadykov, Timur M., editor, and Vorozhtsov, Evgenii V., editor

Published
2020
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13. Implementing the Tangent Graeffe Root Finding Method

Author

van der Hoeven, Joris, Monagan, Michael, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Woeginger, Gerhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Bigatti, Anna Maria, editor, Carette, Jacques, editor, Davenport, James H., editor, Joswig, Michael, editor, and de Wolff, Timo, editor

Published
2020
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14. Some results on counting roots of polynomials and the Sylvester resultant

Author

Monagan, Michael and Tuncer, Baris

Subjects
Mathematics - Combinatorics, Computer Science - Discrete Mathematics, Computer Science - Symbolic Computation
Abstract

We present two results, the first on the distribution of the roots of a polynomial over the ring of integers modulo $n$ and the second on the distribution of the roots of the Sylvester resultant of two multivariate polynomials. The second result has application to polynomial GCD computation and solving polynomial diophantine equations., Comment: Presented at FPSAC 2016, Vancouver, Canada, 2016

Published
2016

15. A Modular Algorithm for Computing Polynomial GCDs over Number Fields presented with Multiple Extensions

Author

van Hoeij, Mark and Monagan, Michael

Subjects
Computer Science - Symbolic Computation
Abstract

We consider the problem of computing the monic gcd of two polynomials over a number field L = Q(alpha_1,...,alpha_n). Langemyr and McCallum have already shown how Brown's modular GCD algorithm for polynomials over Q can be modified to work for Q(alpha) and subsequently, Langemyr extended the algorithm to L[x]. Encarnacion also showed how to use rational number to make the algorithm for Q(alpha) output sensitive, that is, the number of primes used depends on the size of the integers in the gcd and not on bounds based on the input polynomials. Our first contribution is an extension of Encarnacion's modular GCD algorithm to the case n>1, which, like Encarnacion's algorithm, is is output sensitive. Our second contribution is a proof that it is not necessary to test if p divides the discriminant. This simplifies the algorithm; it is correct without this test. Our third contribution is a modification to the algorithm to treat the case of reducible extensions. Such cases arise when solving systems of polynomial equations. Our fourth contribution is an implementation of the modular GCD algorithm in Maple and in Magma. Both implementations use a recursive dense polynomial data structure for representing polynomials over number fields with multiple field extensions. Our fifth contribution is a primitive fraction-free algorithm. This is the best non-modular approach. We present timing comparisons of the Maple and Magma implementations demonstrating various optimizations and comparing them with the monic Euclidan algorithm and our primitive fraction-free algorithm., Comment: 36 pages. An early version of this paper appeared in the Proceedings of ISSAC 2002, ACM Press, pp. 207-213. This version represents work done between 2003 and 2005 outlined as contributions 3, 4 and 5 in the abstract

Published
2016

20. Factoring Multivariate Polynomials with Many Factors and Huge Coefficients

Author

Monagan, Michael, Tuncer, Baris, Hutchison, David, Series Editor, Kanade, Takeo, Series Editor, Kittler, Josef, Series Editor, Kleinberg, Jon M., Series Editor, Mattern, Friedemann, Series Editor, Mitchell, John C., Series Editor, Naor, Moni, Series Editor, Pandu Rangan, C., Series Editor, Steffen, Bernhard, Series Editor, Terzopoulos, Demetri, Series Editor, Tygar, Doug, Series Editor, Weikum, Gerhard, Series Editor, Gerdt, Vladimir P., editor, Koepf, Wolfram, editor, Seiler, Werner M., editor, and Vorozhtsov, Evgenii V., editor

Published
2018
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21. Sparse Multivariate Hensel Lifting: A High-Performance Design and Implementation

Author

Monagan, Michael, Tuncer, Baris, Hutchison, David, Series Editor, Kanade, Takeo, Series Editor, Kittler, Josef, Series Editor, Kleinberg, Jon M., Series Editor, Mattern, Friedemann, Series Editor, Mitchell, John C., Series Editor, Naor, Moni, Series Editor, Pandu Rangan, C., Series Editor, Steffen, Bernhard, Series Editor, Terzopoulos, Demetri, Series Editor, Tygar, Doug, Series Editor, Weikum, Gerhard, Series Editor, Davenport, James H., editor, Kauers, Manuel, editor, Labahn, George, editor, and Urban, Josef, editor

Published
2018
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22. A new edge selection heuristic for computing the Tutte polynomial of an undirected graph

Author

Monagan, Michael

Subjects
Computer Science - Discrete Mathematics, Computer Science - Symbolic Computation, G.2.2
Abstract

We present a new edge selection heuristic and vertex ordering heuristic that together enable one to compute the Tutte polynomial of much larger sparse graphs than was previously doable. As a specific example, we are able to compute the Tutte polynomial of the truncated icosahedron graph using our Maple implementation in under 4 minutes on a single CPU. This compares with a recent result of Haggard, Pearce and Royle whose special purpose C++ software took one week on 150 computers., Comment: 11 pages, 7 figures, 9 tables, to appear in the proceedings of FPSAC 2012

Published
2012

24. Computing Characteristic Polynomials of Matrices of Structured Polynomials

Author

Law, Marshall, Monagan, Michael, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Gerdt, Vladimir P., editor, Koepf, Wolfram, editor, Seiler, Werner M., editor, and Vorozhtsov, Evgenii V., editor

Published
2016
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25. Using Sparse Interpolation in Hensel Lifting

Author

Monagan, Michael, Tuncer, Baris, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Gerdt, Vladimir P., editor, Koepf, Wolfram, editor, Seiler, Werner M., editor, and Vorozhtsov, Evgenii V., editor

Published
2016
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31. Sparse Polynomial Powering Using Heaps

Author

Monagan, Michael, Pearce, Roman, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Gerdt, Vladimir P., editor, Koepf, Wolfram, editor, Mayr, Ernst W., editor, and Vorozhtsov, Evgenii V., editor

Published
2012
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33. Lazy and Forgetful Polynomial Arithmetic and Applications

Author

Monagan, Michael, Vrbik, Paul, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Gerdt, Vladimir P., editor, Mayr, Ernst W., editor, and Vorozhtsov, Evgenii V., editor

Published
2009
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34. Polynomial Division Using Dynamic Arrays, Heaps, and Packed Exponent Vectors

Author

Monagan, Michael, Pearce, Roman, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Ganzha, Victor G., editor, Mayr, Ernst W., editor, and Vorozhtsov, Evgenii V., editor

Published
2007
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38. Global access to Maple information

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., and Watt, Stephen M.

Published
1992
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39. Advanced Examples

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., and Watt, Stephen M.

Published
1992
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40. Measuring and improving performance

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., and Watt, Stephen M.

Published
1992
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41. Advanced Graphics

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., and Watt, Stephen M.

Published
1992
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42. The Maple Programming Language

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., and Watt, Stephen M.

Published
1992
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43. Interactive Use of Maple

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., and Watt, Stephen M.

Published
1992
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44. Less Simple Maple

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton L., Monagan, Michael B., and Watt, Stephen M.

Published
1992
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45. Miscellaneous Packages

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., and Watt, Stephen M.

Published
1991
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46. Packages for Geometry

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., and Watt, Stephen M.

Published
1991
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47. Packages for Applied Mathematics

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., and Watt, Stephen M.

Published
1991
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48. Packages

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., and Watt, Stephen M.

Published
1991
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49. Packages for Discrete Mathematics

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., and Watt, Stephen M.

Published
1991
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50. The Maple Library

Author

Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., Watt, Stephen M., Char, Bruce W., Geddes, Keith O., Gonnet, Gaston H., Leong, Benton, Monagan, Michael B., and Watt, Stephen M.

Published
1991
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