Factoring polynomials with rational coefficients

In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q(X). It is well known that this is equivalent to factoring primitive polynomials feZ(X) into irreducible factors in Z(X). Here we call f~ Z(X) primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. (8). Its running time, measured in bit operations, is O(nl2+n9(log(fD3).

[1]  M. Mignotte An inequality about factors of polynomials , 1974 .

[2]  A. Brentjes,et al.  Multi-dimensional continued fraction algorithms , 1981 .

[3]  Hans Zassenhaus,et al.  A remark on the Hensel factorization method , 1978 .

[4]  A. Odlyzko,et al.  Irreducibility testing and factorization of polynomials , 1983 .

[5]  Donald E. Knuth The Art of Computer Programming 2 / Seminumerical Algorithms , 1971 .

[6]  Leonard M. Adleman,et al.  Irreducibility testing and factorization of polynomials , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[7]  H. Zassenhaus On Hensel factorization, I , 1969 .

[8]  G. Hardy,et al.  An Introduction to the Theory of Numbers , 1938 .

[9]  Arjen K. Lenstra,et al.  Lattices and factorization of polynomials , 1981, SIGS.

[10]  Donald E. Knuth,et al.  The art of computer programming. Vol.2: Seminumerical algorithms , 1981 .

[11]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[12]  Hendrik W. Lenstra,et al.  Integer Programming with a Fixed Number of Variables , 1983, Math. Oper. Res..

[13]  J. Rosser,et al.  Approximate formulas for some functions of prime numbers , 1962 .

[14]  David Y. Y. Yun,et al.  The Hensel Lemma in Algebraic Manipulation , 1973, Outstanding Dissertations in the Computer Sciences.

[15]  Paul Pritchard,et al.  Programming Techniques and Data Structures a Sublinear Additive Sieve for Finding Prime Numbers , 2022 .

[16]  R. Forcade,et al.  Generalization of the Euclidean algorithm for real numbers to all dimensions higher than two , 1979 .

[17]  C. A. Rogers,et al.  An Introduction to the Geometry of Numbers , 1959 .

[18]  David G. Cantor,et al.  Irreducible Polynomials with Integral Coefficients Have Succinct Certificates , 1981, J. Algorithms.