Factorization Of Multivariate Polynomials Research Articles

Factorization of Multivariate Polynomials over the integers

For some time now I have been wondering how well the existing polynomial factorization algorithms perform on some of the more difficult multivariate polynomials. Below are ten (of the over 300) multivariate polynomials that the heuristic program (POLYFACT) with learning factored while I was working on my dissertation (a complete description of POLYFACT will soon appear in the journal <u>Artificial Intelligence)</u>. Table I gives the factorization times for POLYFACT. When any programs factor some or all of the ten polynomials, I think it would be interesting to publish the times (along with the polynomial number(s) in the SIGSAM Bulletin. Please indicate the machine on which the factorization was done. I am submitting the ten polynomials in factored form in order to conserve space; they need to be expanded and simplified prior to factorization attempts.

Factoring multivariate polynomials over algebraic number fields

The algorithm for factoring polynomials over the integers by Wang and Rothschild is generalized to an algorithm for the irreducible factorization of multivariate polynomials over any given algebraic number field. The extended method makes use of recent ideas in factoring univariate polynomials over large finite fields due to Berlekamp and Zassenhaus. The procedure described has been implemented in the algebraic manipulation system MACSYMA. ∗ ∗ ^{\ast \ast } Some machine examples with timing are included.

A new approach to the symbolic factorization of multivariate polynomials

A heuristic factorization scheme that uses learning and other heuristic programming techniques to improve the efficiency of determining the symbolic factorization of multivariate polynomials with integer coefficients and an arbitrary number of variables and terms is described. The learning program, POLYFACT, in which the factorization scheme is implemented is also described. POLYFACT uses learning through the dynamic construction and manipulation of first-order predicate calculus heuristics to reduce the amount of searching for the irreducible factors of a polynomial. Tables containing the results of factoring randomly generated multivariate polynomials are presented: (1) to demonstrate that learning does improve considerably the efficiency of factoring polynomials, and (2) to show that POLYFACT does learn from previous experience. The factorization times of polynomials factored by both the scheme implemented in POLYFACT and Wang's implementation of Berlekamp's algorithm are given. The two algorithms are compared, and two situations where POLYFACT'S algorithm can be used to improve the efficiency of Wang's algorithm are discussed.

Factoring multivariate polynomials over algebraic number fields in MACSYMA

The algorithm for factoring polynomials over the integers by Wang and Rothschild is generalized to an algorithm for the irreducible factorization of multivariate polynomials over any given algebraic number field. The extended method makes use of recent ideas in factoring univariate polynomials over large finite fields due to Berlekamp and Zassenhaus. The procedure has been implemented in the algebraic manipulation system MACSYMA. Some machine examples with timing are presented.

Multivariate Polynomial Factorization

Abstract : This paper describes algorithms for factoring a polynomial in one or more variables, with integer coefficients, into factors which are irreducible over the integers. These algorithms are based on the use of factorizations over finite fields and 'Hensel's Lemma construction'. 'Abstract algorithm' descriptions are used in the presentation of the underlying algebraic theory. Included is a new generalization of Hensel's p-adic construction which leads to a practical algorithm for factoring multivariate polynomials. The univariate case algorithm is also specified in greater detail than in the previous literature, with attention to a number of improvements which the author has developed based on theoretical computing time analyses and experience with actual implementations.