# Factoring polynomials via relation-finding

## Abstract

In this paper we describe a new algorithm for fully factoring polynomials defined over the rationals or over number-fields. This algorithm uses as an essential subroutine, any fast relation finding algorithm for vectors of real numbers. Unlike previous algorithms which work on one factor at a time, the new algorithm finds all factors at once. Let *P* be a polynomial of degree *n*, height *H(P*) (=sum of the absolute values of *P*'s coefficients), logarthmic height *h(P)*=log *H(P)*. If we use the HJLS relation-finding algorithm of Hastad, Just, Lagarias and Schnorr, our algorithm has running time O(*n*^{5+C}h(P)) if fast multiplication is used, and O(*n*^{6+C}h(P)) if ordinary multiplication is used. This is an improvement by a factor of *n* over the algorithm of Schönhage, the previously best known.

## Preview

Unable to display preview. Download preview PDF.

## References

- [BF89]David H. Bailey and Helaman R. P. Ferguson. Numerical results on relations between fundamental constants using a new algorithm.
*Math. Comp.*, 53:649–656, 1989.Google Scholar - [FB91]Helaman R. P. Ferguson and David H. Bailey. A polynomial time, numerically stable integer relation algorithm. Technical report, Supercomputing Research Center, 17100 Science Drive, Bowie, MD 20715, December 1991.Google Scholar
- [GMT89]Patrizia Gianni, Victor Miller, and Barry Trager. Decomposition of algebras. In
*ISAAC '88*, Lecture Notes in Computer Science. Springer-Verlag, 1989.Google Scholar - [HJLS89]Johann Hastad, B. Just, Jeffrey C. Lagarias, and Claus P. Schnorr. Polynomial time algorithms for finding integer relations among real numbers.
*Siam J. Comput.*, 18:859–881, 1989.Google Scholar - [KM86]Ravi Kannan and Lyle A. McGeoch. Basis reduction and evidence for transcendence of certain numbers. In
*Foundations of software technology and theoretical computer science (New Delhi, 1986)*, volume 241 of*Lecture Notes in Comput. Sci.*, pages 263–269. Springer, Berlin-New York, 1986.Google Scholar - [LLL82]Arjen K. Lenstra, Hendrik W. Lenstra, Jr., and Laszlo Lovasz. Factoring polynomials with rational coefficients.
*Mathematische Annalen*, 261:515–534, 1982.Google Scholar - [Mah64]Kurt Mahler. An inequality for the discriminant of a polynomial.
*Michigan Math. J.*, 11:257–262, 1964Google Scholar - [Sch82]Arnold Schönhage. The fundamental theorem of algebra in terms of computational complexity. Technical report, Math. Inst. Univ. Tübingen, 1982.Google Scholar
- [Sch84]Arnold Schönhage. Factorization of univariate integer polynomials by diophantine approximation and an improved basis reduction algorithm. In
*Automata, languages and programming (Antwerp, 1984)*, volume 172 of*Lecture Notes in Computer Science*, pages 436–447, Berlin-New York, 1984. Springer-Verlag.Google Scholar - [Sch87]Arnold Schönhage. Equation solving in terms of computational complexity. In
*Proceedings of the International Congress of Mathematicians, 1986*, volume 1, pages 131–153. American Mathematical Society, 1987.Google Scholar