Efficient multivariate factorization over finite fields
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 )[x1, x2,..., x v ]. We also needed an efficient implementation of our algorithms in ℤ p [y][x] 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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- 1.Bernardin, L. Fast dense Hensel lifting, manuscript, ETH Zürich, 1995. Available via http://www.inf.ethz.ch/personal/bernardi.Google Scholar
- 2.Bernardin, L. On square-free factorization of multivariate polynomials over a finite field. Theoretical Computer Science 187 (1997). to appear.Google Scholar
- 3.Cantor, D. G., and Zassenhaus, H. A new algorithm for factoring polynomials over finite fields. Mathematics of Computation 36, 154 (1981), 587–592.Google Scholar
- 4.Czapor, S. R. Solving algebraic equations:combining Buchberger's algorithm with multivariate factorization. Journal of Symbolic Computation 7, 1 (January 1989), 49–54.Google Scholar
- 5.da Rosa, R. M. Private communication, February 1996.Google Scholar
- 6.Geddes, K. O., Czapor, S. R., and Labahn, G.Algorithms for Computer Algebra. Kluwer Academic Publishers, Boston, 1992.Google Scholar
- 7.Jenks, R., and Sutor, R. AXIOM: The Scientific Computation System. Springer Verlag, 1992.Google Scholar
- 8.Kaltofen, E. Sparse Hensel lifting. In Proceedings of Eurocal '85, Vol. II (1985), B. F. Caviness, Ed., vol. 204 of Lecture Notes in Computer Science, Springer-Verlag, pp. 4–17.Google Scholar
- 9.Kaltofen, E., Musser, D. R., and Saunders, B. D. A generalized class of polynomials that are hard to factor. SIAM Journal on Computing 12, 3 (1983), 473–485.Google Scholar
- 10.Kaltofen, E., and Trager, B. M. Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators. Journal of Symbolic Computation 9, 3 (March 1990), 300–320.Google Scholar
- 11.Knuth, D. E. Seminumerical Algorithms, vol. 2 of The Art of Computer Programming. Addison Wesley, 1981.Google Scholar
- 12.Lucks, M. A fast implementation of polynomial factorization. In SYMSAC '86: Proceedings of the 1986 ACM Symposium on Symbolic and Algebraic Computation (1986), pp. 228–232.Google Scholar
- 13.Monagan, M. B. Gauss: A parameterized domain of computation system with support for signature functions. In Proceedings of DISCO '93 (1993), vol. 722 of Lecture Notes in Computer Science, Springer-Verlag, pp. 81–94.Google Scholar
- 14.Monagan, M. B. In-place arithmetic for polynomials over Zn. In Proceedings of DISCO '92 (1993), vol. 721 of Lecture Notes in Computer Science, Springer-Verlag, pp. 22–34.Google Scholar
- 15.Popp, H. Moduli Theory and Classification Theory of Algebraic Varieties, vol. 620 of Lecture Notes in Mathematics. Springer-Verlag, 1977.Google Scholar
- 16.Shoup, V. A new polynomial factorization algorithm and its implementation. Journal of Symbolic Computation 20, 4 (1995), 363–397.Google Scholar
- 17.Stoutmyer, D. R. Which polynomial representation is best? In Proceedings of the 1984 MACSYMA User's Conference (1984), V. E. Golden, Ed., pp. 221–243.Google Scholar
- 18.Swanson, S. L. On the Factorization of Multivariate Polynomials over Finite Fields. PhD thesis, Purdue University, 1993.Google Scholar
- 19.von Zur Gathen, J. Factoring sparse multivariate polynomials. In Proceedings of the 24th IEEE Symposium on Foundations of Computer Science (1983), pp. 172–179.Google Scholar
- 20.von Zur Gathen, J. Irreducibility of multivariate polynomials. Journal of Computer and System Sciences 31 (1985), 225–264.Google Scholar
- 21.von Zur Gathen, J., and Kaltofen, E. Polynomial-time factorization of multivariate polynomials over finite fields. In Proceedings of ICALP '83 (1983), vol. 154 of Lecture Notes in Computer Science, Springer-Verlag, pp. 250–262.Google Scholar
- 22.von Zur Gathen, J., and Kaltofen, E. Factoring sparse multivariate polynomials. Journal of Computer and System Sciences 31 (1985), 265–287.Google Scholar
- 23.Waterloo Maple Inc. Maple V learning guide. Springer-Verlag, 1996.Google Scholar