Factorization of multivariate polynomials over finite fields

Japan Journal of Industrial and Applied Mathematics, 1993

Recently, Sasaki et al. presented an approximate factorization algorithm of multivariate polynomials. The algorithm calculates irreducible factors by investigating linear combinations of the same power of appraximate roots. In this paper, we show that various kinds of multivaxiate polynomial factorizations can be performed by this method. We present algorithms for factorization of multivaxiate polynomials over power-series rings, over the integers, over algebralc number fields including algebraically closed fields, and over algebraic function fields. Furthermore, we discuss applicability of this method to univariate polynomial factorization.

Theoretical Computer Science, 2000

We analyse an extension of Shoup's (Inform. Process. Lett. 33 (1990) 261-267) deterministic algorithm for factoring polynomials over ÿnite prime ÿelds to arbitrary ÿnite ÿelds. In particular, we prove the existence of a deterministic algorithm which completely factors all monic polynomials of degree n over F q; q odd, except possibly O(n 2 log 2 q=q) polynomials, using O(n 2+ log 2 q) arithmetical operations in Fq.

Journal of Symbolic Computation, 2005

In this paper we describe software for an efficient factorization of polynomials over global fields F. The algorithm for function fields was recently incorporated into our system KANT. The method is based on a generic algorithm developed by the second author in an earlier paper in this journal. Besides algorithmic aspects not contained in that paper we give details about the current implementation and about some complexity issues as well as a few illustrative examples. Also, a generalization of the application of LLL reduction for factoring polynomials over arbitrary global fields is developed.

Theoretical Computer Science, 2011

Shuhong Gao (2003) [6] has proposed an efficient algorithm to factor a bivariate polynomial f over a field F. This algorithm is based on a simple partial differential equation and depends on a crucial fact: the dimension of the polynomial solution space G associated with this differential equation is equal to the number r of absolutely irreducible factors of f. However, this holds only when the characteristic of F is either zero or sufficiently large in terms of the degree of f. In this paper we characterize a vector subspace of G for which the dimension is r, regardless of the characteristic of F, and the properties of Gao's construction hold. Moreover, we identify a second vector subspace of G that leads to an analogous theory for the rational factorization of f .

Proceedings of the 2004 international symposium on Symbolic and algebraic computation - ISSAC '04, 2004

Many polynomial factorization algorithms rely on Hensel lifting and factor recombination. For bivariate polynomials we show that lifting the factors up to a precision linear in the total degree of the polynomial to be factored is sufficient to deduce the recombination by linear algebra, using trace recombination. Then, the total cost of the lifting and the recombination stage is subquadratic in the size of the dense representation of the input polynomial. Lifting is often the practical bottleneck of this method: we propose an algorithm based on a faster multi-moduli computation for univariate polynomials and show that it saves a constant factor compared to the classical multifactor lifting algorithm.

Journal of Symbolic Computation, 1996

The paper describes improved techniques for factoring univariate polynomials over the integers. The authors modify the usual linear method for lifting modular polynomial factorizations so that efficient early factor detection can be performed. The new lifting method is universally faster than the classical quadratic method, and is faster than a linear method due to Wang, provided we lift sufficiently high. Early factor detection is made more effective by also testing combinations of modular factors, rather than just single modular factors. Various heuristics are presented that reduce the cost of the factor testing or that increase the chance of successful testing. Both theoretical and empirical computing times are presented.

Journal of Symbolic Computation, 2005

In this paper we present a generic algorithm for factoring polynomials over global fields F. As efficient implementations of that algorithm for number fields and function fields differ substantially, these cases will be treated separately. Complexity issues and implementations will be discussed in part II which also contains illustrative examples.

State of the art factoring in Q[x] is dominated in theory by a combinatorial reconstruction problem while, excluding some rare polynomials, performance tends to be dominated by Hensel lifting. We present an algorithm which gives a practical improvement (less Hensel lifting) for these more common polynomials. In addition, factoring has suffered from a 25 year complexity gap because the best implementations are much faster in practice than their complexity bounds. We illustrate that this complexity gap can be closed by providing an implementation which is comparable to the best current implementations and for which competitive complexity results can be proved.

2018

Our goal is to develop a high-performance code for factoring a multivariate polynomial in n variables with integer coefficients which is polynomial time in the sparse case and efficient in the dense case. Maple, Magma, Macsyma, Singular and Mathematica all implement Wang’s multivariate Hensel lifting, which, for sparse polynomials, can be exponential in n. Wang’s algorithm is also highly sequential. In this work we reorganize multivariate Hensel lifting to facilitate a highperformance parallel implementation. We identify multivariate polynomial evaluation and bivariate Hensel lifting as two core components. We have also developed a library of algorithms for polynomial arithmetic which allow us to assign each core an independent task with all the memory it needs in advance so that memory management is eliminated and all important operations operate on dense arrays of 64 bit integers. We have implemented our algorithm and library using Cilk C for the case of two monic factors. We disc...

Advances in Applied Mathematics, 2010

This paper presents a new algorithm for the absolute factorization of parametric multivariate polynomials over the field of rational numbers. This algorithm decomposes the parameters space into a finite number of constructible sets. The absolutely irreducible factors of the input parametric polynomial are given uniformly in each constructible set. The algorithm is based on a parametric version of Hensel's lemma and an algorithm for quantifier elimination in the theory of algebraically closed field in order to reduce the problem of finding absolute irreducible factors to that of representing solutions of zero-dimensional parametric polynomial systems. The complexity of this algorithm is single exponential in the number n of the variables of the input polynomial, its degree d w.r.t. these variables and the number r of the parameters.

Journal de Théorie des Nombres de Bordeaux, 2009

We prove polynomial time complexity for a now widely used factorization algorithm for polynomials over the rationals. Our approach also yields polynomial time complexity results for bivariate polynomials over a finite field.

SIAM Journal on Computing, 1983

It is shown that any multivariate polynomial of degree d that can be computed sequentially in C steps can be computed in parallel in O((log d)(log C + log d)) steps using only (Cd) 1) processors.

Discrete Mathematics, 2019

Let F q be the finite field with q elements, where q is a prime power and n be a positive integer. In this paper, we explore the factorization of f (x n) over F q , where f (x) is an irreducible polynomial over F q. Our main results provide generalizations of recent works on the factorization of binomials x n − 1. As an application, we provide an explicit formula for the number of irreducible factors of f (x n) under some generic conditions on f and n.

Journal of Complexity, 2007

We present a deterministic algorithm for computing all irreducible factors of degree ≤ d of a given bivariate polynomial f ∈ K[x, y] over an algebraic number field K and their multiplicities, whose running time is polynomial in the bit length of the sparse encoding of the input and in d . Moreover, we show that the factors over Q of degree ≤ d which are not binomials can also be computed in time polynomial in the sparse length of the input and in d .

Ukrainian Mathematical Journal, 1999

We propose and justify a numerical method of factorization of polynomials with complex coefficients. We construct an algorithm of factorization of polynomials with real coefficients into real factors in the case of multiple roots. We propose and justify an algorithm of factorization of polynomials with complex coefficients. In a special case, we consider the factorization of a polynomial with real coefficients into real factors.