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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Factoring multivariate polynomials over the integers
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by Paul S. Wang and Linda Preiss Rothschild PDF
Math. Comp. 29 (1975), 935-950 Request permission

Abstract:

An algorithm for the irreducible factorization of any multivariate polynomial over the integers is given. It is much faster than the classical method ascribed to Kronecker. The algorithm begins by making substitutions for all but one of the variables with selected integers, giving a polynomial in just one variable. This univariate polynomial is then factored by a known method, which uses an algorithm of Berlekamp for factoring univariate polynomials over finite fields. The multivariate factors are constructed from the univariate ones by a kind of Hensel algorithm. The procedure has been implemented in the algebraic manipulation systems MACSYMA and SCRATCHPAD. A number of machine examples with timing are included.
References
  • E. R. Berlekamp, Factoring polynomials over finite fields, Bell System Tech. J. 46 (1967), 1853–1859. MR 219231, DOI 10.1002/j.1538-7305.1967.tb03174.x
  • E. R. Berlekamp, Factoring polynomials over large finite fields, Math. Comp. 24 (1970), 713–735. MR 276200, DOI 10.1090/S0025-5718-1970-0276200-X
  • W. S. Brown, On Euclid’s algorithm and the computation of polynomial greatest common divisors, J. Assoc. Comput. Mach. 18 (1971), 478–504. MR 307450, DOI 10.1145/321662.321664
    • G. E. COLLINS, SAC-1 Modular Arithmetic System, University of Wisconsin Technical Report No. 10, June 1969.
  • A. O. Gel′fond, Transcendental and algebraic numbers, Dover Publications, Inc., New York, 1960. Translated from the first Russian edition by Leo F. Boron. MR 0111736
  • Donald E. Knuth, The art of computer programming. Vol. 2: Seminumerical algorithms, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0286318
    • J. McCARTHY et al., LISP 1.5 Programmer’s Manual, M.I.T. Press, Cambridge, Mass., 1963. J. MOSES & D. Y. Y. YUN, "The EZ GCD algorithm," Proceedings of ACM Annual Conference, August 1973. D. R. MUSSER, Algorithms for Polynomial Factorization, Ph. D. Thesis, Computer Science Department, The University of Wisconsin, Madison, Wis., 1971. B. L. VAN DER WAERDEN, Modern Algebra. Vol. 1, Springer, Berlin, 1930; English transl., Ungar, New York, 1949. MR 10, 587. D. Y. Y. YUN, The Hensel Lemma in Algebraic Manipulation, Ph. D. Thesis, Department of Mathematics, M.I.T., Nov. 1973 (also Project MAC TR-138, November 1974).
  • Hans Zassenhaus, On Hensel factorization. I, J. Number Theory 1 (1969), 291–311. MR 242793, DOI 10.1016/0022-314X(69)90047-X
    • MACSYMA Reference Manual, the MATHLAB group, Project MAC, M.I.T., Cambridge, Mass., Sept. 1974.
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Math. Comp. 29 (1975), 935-950
  • MSC: Primary 10M05; Secondary 68A10
  • DOI: https://doi.org/10.1090/S0025-5718-1975-0396471-3
  • MathSciNet review: 0396471