Factoring univariate integral polynomials in polynomial average time
Let A be a primitive squarefree univariate integral polynomial of degree n. An irreducible factor of A can be found by forming products of lifted modulo p factors of A for a suitable small prime p. One can either form first the products consisting of the smallest numbers of lifted factors (cardinality procedure) or form first the products with smallest degrees (degree procedure). Let ∏ be the partition of n consisting of the degrees of the irreducible factors of A. The average number of products formed before finding an irreducible factor of A is a function of ∏, C(∏) or D(∏) respectively. Let C*(n) (D*(n)) be the maximum of C(∏) (D(∏)) for all partitions, ∏, of n. Subject to the validity of two conjectures, for which considerable evidence is presented, it is proved that C*(n) is dominated by n2 whereas D*(n) is exponential. If the conjectures are true then the cardinality procedure results in a complete factorization algorithm for primitive univariate integral polynomials whose average computing time, in a very strong sense, is dominated by a polynomial function of its degree n.
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- Berlekamp, E. R. Algebraic Coding Theory. McGraw-Hill, 1968.Google Scholar
- Collins, G.E., and Musser, D.R. The SAC-l Polynomial Factorization System. Technical Report #157, Computer Sciences Department, Univ. of Wisconsin-Madison, March 1972.Google Scholar
- Frobenius, F.G. Über Beziehungen zwischen den Primidealen eines algebraischen Körpers und den Substitutionen seiner Gruppe. (1896). Gesammelte Abhandlungen II.Google Scholar
- Janusz, G.J. Algebraic Number Fields. Academic Press, 1973.Google Scholar
- Knuth, D.E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms. Addison-Wesley, 1968.Google Scholar
- Knuth, D.E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms. Addison-Wesley, 1968.Google Scholar
- Musser, D.R. Algorithms for Polynomial Factorization. (Ph.D. thesis). Technical Report #134, Computer Sciences Department, Univ. of Wisconsin-Madison, Sept. 1974.Google Scholar
- Musser, D.R. Multivariate Polynomial Factorization. Jour. ACM, Vol. 22, No. 2 (April 1975), pp. 291–308.Google Scholar
- Musser, D.R. On the Efficiency of a Polynomial Irreducibility Test. Jour. ACM, Vol. 25, No. 2 (April 1978), pp. 271–282.Google Scholar
- van der Waerden, B.L. Die Seltenheit der Gleichungen mit Affekt. Math. Ann. 109 (1936), pp. 13–16.Google Scholar
- van der Waerden, B.L. Modern Algebra, Vol. 1. Ungar, 1948.Google Scholar