Abstract
The problem of factorising polynomials: that is to say, given a polynomial with integer coefficients, to find the irreducible polynomials that divide it, is one with a long history. While the last word has not been said on the subject, we can say that the past 15 years have seen major break-throughs, and many computer algebra systems now include efficient algorithms for this problem. When it comes to polynomials with algebraic number coefficients, the problem is far harder, and several major questions remain to be answered. Nevertheless, the last few years have seen substantial improvements, and such factorisations are now possible.
This research is partially supported by the Science and Engineering Research Council, and the University of Bath Research Fund
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References
Abbott,J.A. & Davenport,J.H., A Remark on a Paper by Wang: Another Surprising Property of 42. Submitted to Math. Comp.
[Abbott et al., 1985] Abbott, J.A., Bradford, R.J. & Davenport, J.H., A Remark on Factorisation. SIGSAM Bulletin 19 (1985) 2, pp. 31–33, 37.
[Abbott et al., 1986a] Abbott, J.A., Bradford, R.J. & Davenport, J.H., The Bath Algebraic Number Package. Proc. SYMSAC 86 (ACM, New York, 1986) pp. 250–253.
Berlekamp, E.R., Factoring Polynomials over Finite Fields. Bell System Tech. J. 46 (1967) pp. 1853–1859.
Berlekamp, E.R., Factoring Polynomials over Large Finite Fields. Math. Comp. 24 (1970) pp. 713–735.
Berwick,W.E.H., Integral Bases. Cambridge Tracts in Mathematics and Mathematical Physics 22, C.U.P., 1928.
Bradford, R.J., On the Computation of Integral Bases. PhD. Thesis, University of Bath.
Böffgen, R., Private communication. June 1987
Cantor, D.G. & Zassenhaus, H., A New Algorithm for Factoring Polynomials over Finite Fields. Math. Comp. 36 (1981) pp. 587–592. Zbl. 493.12024. MR 82e:12020.
Collins, G.E., Subresultants and Reduced Polynomial Remainder Sequences. J. ACM 14 (1967) pp. 128–142.
Collins, G.E., Factoring univariate integral polynomials in polynomial average time. Proc. EUROSAM 79 (Springer Lecture Notes in Computer Science 72, Springer-Verlag, Berlin-Heidelberg-New York) pp. 317–329. MR 81g:68064.
Ford,D.J., On the Computation of the Maximal order in a Dedekind Domain. Ph.D. Thesis, Ohio State University, 1978.
Kaltofen,E., Sparse Hensel Lifting. Proc. EUROCAL 85, Vol. 2 (Springer Lecture Notes in Computer Science Vol. 204, Springer-Verlag, 1985) pp. 4–17.
[Kaltofen et al., 1981] Kaltofen, E., Musser, D.R. & Saunders, B.D., A Generalized Class of Polynomials That are Hard to Factor. Proc. SYMSAC 81 (ACM, New York, 1981) pp. 188–194. Zbl. 477.68041.
[Kaltofen et al., 1983] Kaltofen, E., Musser, D.R. & Saunders, B.D., A Generalized Class of Polynomials That are Hard to Factor. SIAM J. Comp. 12 (1983) pp. 473–483. CR 8405-0367 (Vol. 25 (1984) p. 235). MR 85a:12001.
Landau, E., Sur Quelques Théorèmes de M. Petrovic Relatif aux Zéros des Fonctions Analytiques. Bull. Soc. Math. France 33 (1905) pp. 251–261.
Lenstra, A.K., Lattices and Factorization of Polynomials over Algebraic Number Fields. Proc. EUROCAM 82 [Springer Lecture Notes in Computer Science 144, Springer-Verlag, Berlin-Heidelberg-New York, 1982], pp. 32–39. Zbl. 495.68005.
Lenstra, A.K., Factoring Polynomials over Algebraic Number Fields. Proc. EUROCAL 83 [Springer Lecture Notes in Computer Science 162, Springer-Verlag, Berlin-Heidelberg-New York, 1983] pp. 245–254.
Lenstra, A.K., Factoring Multivariate Polynomials over Algebraic Number Fields. SIAM J. Comp. 16 (1987) pp. 591–598.
[Lenstra et al., 1982] Lenstra, A.K., Lenstra, H.W., Jun. & Lovász, L., Factoring Polynomials with Rational Coefficients. Math. Ann. 261 (1982) pp. 515–534. Zbl. 488.12001. MR 84a: 12002.
Mignotte, M., An Inequality about Factors of Polynomials. Math. Comp. 28 (1974) pp. 1153–1157. Zbl. 299.12101.
Mignotte, M., Some Useful Bounds. Symbolic & Algebraic Computation (Computing Supplementum 4) (ed. B. Buchberger, G.E. Collins & R. Loos) Springer-Verlag, Wien-New York, 1982, pp. 259–263. Zbl. 498.12019.
Musser, D.R., On the efficiency of a polynomial irreducibility test. J. ACM 25 (1978) pp. 271–282. MR 80m:68040.
Trager, B.M., Algebraic Factoring and Rational Function Integration. Proc. SYMSAC 76 (ACM, New York, 1976) pp. 219–226. Zbl. 498.12005.
Trager,B.M., Private Communication. April 1987.
Vaughan, T.P., On Computing the Discriminant of an Algebraic Number Field. Math. Comp. 45 (1985) pp. 569–584.
Wang, P.S., Factoring Multivariate Polynomials over Algebraic Number Fields. Math. Comp. 30 (1976) pp. 324–336.
[Wang et al., 1982] Wang, P.S., Guy, M.J.T. and Davenport, J.H., P-adic Reconstruction of Rational Numbers. SIGSAM Bulletin 16 (1982) 2 pp. 2–3. Zbl. 489.68032.
Weinberger, P.J. & Rothschild, L.P., Factoring Polynomials over Algebraic Number Fields. ACM Transactions on Mathematical Software 2 (1976) pp. 335–350.
Weyl,H., Algebraic Theory of Numbers. Annals of Mathematics Studies 1, Princeton U.P., 1940.
Zassenhaus, H., On the Second Round of the Maximal Order Program. Applications of Number Theory to Numerical Analysis (ed. S.K. Zaremba), Academic Press, New York-London, 1972, pp. 389–431.
Zassenhaus, H., On Hensel Factorization. II. Symposia Mathematica 15 (1975) pp. 499–513.
Zassenhaus, H., On Structural Stability. Communications in Algebra 8 (1980) pp. 1799–1844.
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Abbott, J.A., Bradford, R.J., Davenport, J.H. (1988). Factorisation of polynomials: Old ideas and recent results. In: Janßen, R. (eds) Trends in Computer Algebra. Lecture Notes in Computer Science, vol 296. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-18928-9_6
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