Factorisation of Polynominals: Old Ideas and Recent Results

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.

[1]  Barry M. Trager,et al.  Algebraic factoring and rational function integration , 1976, SYMSAC '76.

[2]  Arjen K. Lenstra,et al.  Lattices and Factorization of Polynomials over Algebraic Number Fields , 1982, EUROCAM.

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

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

[5]  James H. Davenport,et al.  P-adic reconstruction of rational numbers , 1982, SIGS.

[6]  David R. Musser,et al.  On the Efficiency of a Polynomial Irreducibility Test , 1978, JACM.

[7]  H. Zassenhaus.  On structural stability , 1980 .

[8]  James H. Davenport,et al.  The Bath algebraic number package , 1986, SYMSAC '86.

[9]  George E. Collins,et al.  Subresultants and Reduced Polynomial Remainder Sequences , 1967, JACM.

[10]  Erich Kaltofen.  Sparse Hensel Lifting , 1985, European Conference on Computer Algebra.

[11]  James H. Davenport,et al.  A remark on factorisation , 1985, SIGS.

[12]  M. Mignotte.  Some Useful Bounds , 1983 .

[13]  Arjen K. Lenstra.  Factoring Multivariate Polynomials over Algebraic Number Fields , 1987, SIAM J. Comput..

[14]  E. Landau,et al.  Sur quelques théorèmes de M. Petrovitch relatifs aux zéros des fonctions analytiques , 1905 .

[15]  E. Berlekamp.  Factoring polynomials over finite fields , 1967 .

[16]  Peter J. Weinberger,et al.  Factoring Polynomials Over Algebraic Number Fields , 1976, TOMS.

[17]  James H. Davenport,et al.  A remark on a paper by Wang: another surprising property of 42 , 1988 .

[18]  George E. Collins.  Factoring univariate integral polynomial in polynomial average time , 1979, EUROSAM.

[19]  Erich Kaltofen,et al.  A Generalized Class of Polynomials that are Hard to Factor , 1983, SIAM J. Comput..

[20]  Paul S. Wang.  Factoring multivariate polynomials over algebraic number fields , 1976 .

[21]  David James Ford.  On the computation of the maximal order in a dedekind domain. , 1978 .

[22]  Arjen K. Lenstra,et al.  Factoring polynominals over algebraic number fields , 1983, EUROCAL.

[23]  D. Cantor,et al.  A new algorithm for factoring polynomials over finite fields , 1981 .

[24]  Hermann Weyl,et al.  Algebraic Theory of Numbers , 1940 .

[25]  E. Berlekamp.  Factoring polynomials over large finite fields* , 1970, SYMSAC '71.

[26]  On computing the discriminant of an algebraic number field , 1985 .

[27]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .