A pre-test for factoring bivariate polynomials with coefficients in F2

https://doi.org/10.1016/j.ipl.2017.01.009Get rights and content

Highlights

  • We find factors of bivariate polynomials with coefficients in F2.

  • We combine the basic structure of well-known algorithms by Gao and Lecerf.

  • Computational experiments suggest that our procedure speeds up this factorization on Maple.

Abstract

We introduce a pre-test for bivariate polynomial factorization over F2, which combines the basic structure of an algorithm due to Lecerf (2010) [14] with ideas of Gao (2003) [5].

Introduction

This paper considers the factorization of bivariate polynomials whose coefficients lie in the field F2, that is, the aim is to write a polynomial fF2[x,y] as a product f=f1fr of irreducible factors.

The factorization of bivariate polynomials lies at the heart of the factorization of multivariate polynomials with coefficients in a field F. The typical strategy to factor a polynomial fF[x1,,xn] is to perform a reduction, that is, to evaluate some of the indeterminates at elements of F, or replace them by polynomials in other indeterminates, in such a way that the resulting polynomial f˜ has a smaller number of indeterminates. The reduced polynomial f˜ is then factored in the hope that its factorization unveils the factorization of the original polynomial. This strategy works particularly well when the reduction preserves the factorization pattern of the multivariate polynomial f, that is, when each irreducible factor of f is mapped to an irreducible polynomial, and distinct factors are mapped to distinct polynomials. In this case, the factorization of f may be directly recovered from the factors of the reduced polynomial using a method known as Hensel lifting [8], which has been widely used in modern factorization algorithms since the work of Zassenhaus [18] involving polynomials over Q. As it turns out, there are several substitutions from the multivariate to the bivariate setting for which a random selection of parameters preserves the factorization pattern with large probability, provided that the coefficients lie in a sufficiently large field (see, for instance, von zur Gathen [6], Kaltofen [10], [11], Gao [5], Lecerf [13], and Allem, Gao and Trevisan [1]). However, there are no such probabilistic bounds for substitutions to the univariate case. For an in-depth discussion of multivariate factorization, we refer the reader to von zur Gathen and Gerhard [7] and to Mullen and Panario [16], and the references therein. A historical perspective on the evolution of lifting and recombination techniques in polynomial factorization may be found in Lecerf [12].

This has motivated the design of several algorithms for factoring bivariate polynomials. In this paper, we shall be particularly interested in algorithms devised by Gao [5] and Lecerf [12], [14]. These algorithms employ linear algebra tools and were influenced by earlier algorithms for univariate polynomials due to Berlekamp [2], [3] and Niederreiter [15].

The algorithm of Gao [5] is based on the polynomial solutions of a partial differential equation, which had been considered by Ruppert [17] to decide whether a bivariate polynomial is irreducible. Solving this differential equation for polynomials under some particular degree constraints amounts to solving a system of linear equations, and the crucial point in Gao's factorization is that any basis of the solution space may be used to find the factorization of the input polynomial fF[x,y] by performing univariate factorizations, provided that the characteristic of the field F is zero or sufficiently large. As a matter of fact, this restriction on the characteristic of F implies that, with reasonable probability, a single univariate factorization suffices. Rodrigues, Trevisan and the second author [9] have extended Gao's method to small finite fields by identifying an appropriate subspace of this solution space to which Gao's original methods may be applied. They also presented a method to compute this subspace, which unfortunately incurs a significant loss in efficiency unless F=F2.

Lecerf's approach [12], [14] uses the natural reduction strategy of randomly choosing an element cF and factoring the univariate polynomial f(c,y). Unlike previous algorithms using this reduction, the efficiency of his procedure does not depend on the factorization pattern being preserved; it suffices that f(c,y) remains squarefree. Using the classical Hensel lifting strategy, this univariate factorization leads to a decomposition of f into a product of elements G1,,Gs in the ring of formal power series F[[x]][y], which are computed to a certain finite precision. The irreducible factors of f in F[x,y] are obtained through the reduced echelon basis of a linear system derived from G1,,Gs.

Even though Lecerf's approach does not require a lower bound on the characteristic of the field F, the restriction that f(c,y) remains squarefree after the substitution x=c is quite strong if the field is small. For example, since there are only two irreducible polynomials of degree one and one irreducible polynomial of degree two over F2, the restriction f(c,y) will never be squarefree if f(x,y) has two irreducible factors of degree two and one irreducible factor of degree 1, for instance. Because of this, bivariate polynomials with coefficients in F2 are typically factored over extension fields.

In this work, we introduce an algorithm for polynomials over F2 for which all computations in the reduction and recovery steps are done in F2. It follows the main steps of the above method, but does not require f(x,c) to be squarefree. The price to pay is that the factors obtained may be reducible. Given a polynomial fF2[x,y], we make a substitution y=c for some cF2, obtain the univariate factorizationf(x,c)=F1(x)e1Fs(x)es and extend it to polynomials G1,,Gs in the ring of formal power series F2[[y]][x]. These are computed to the same finite precision as in Lecerf's strategy, giving rise to polynomials G1,,GsF2[x,y]. To obtain a factorization of f, we turn to Gao's approach and define a vector space Vˆ over F2 which is derived from G1,,Gs. A factorization of f is obtained through a suitable subspace V of Vˆ, whose elements satisfy precisely the condition imposed in [9] to extend Gao's algorithm to fields of small characteristic. A factorization of f may be obtained from any basis of V through gcd computations, and the number of factors is at least as large as the dimension of V.

The obvious shortcoming of the algorithm in this paper is that the factors obtained are not necessarily irreducible. In the worse case, we would have dim(V)=1, so that no proper factors of f would be obtained. However, we provide a theoretical lower bound on the number of factors found by our algorithm, which is related with the number of classes of irreducible factors of f(x,y) that do not “merge” with each other when the reduction is performed.

The remainder of the paper consists of four sections. In Section 2, we introduce the vector spaces that will be used in this paper and present our main theoretical result, which is proved for polynomials over a general perfect field F. In the particular case of F=F2, this result motivates the algorithm of Section 3. Section 4 reports the results of some computational experiments run on Maple 15, while Section 5 includes some final remarks and open problems.

Section snippets

Main technical results

Even though our main contribution lies in the design of an algorithm that factors bivariate polynomials over F2, most of the theoretical tools that justify its validity may be stated for polynomials over an arbitrary perfect field F. Let fF[x,y] be a polynomial with bidegree (m,n), that is, with degree m with respect to the indeterminate x and n with respect to the indeterminate y. Further suppose that the leading term of f is xm and that f is squarefree, which corresponds to the normalized

The algorithm

In this section, we shall present an efficient algorithm which, for any fixed cF2, finds a factorization of a squarefree bivariate polynomial fF2[x,y] into at least s factors, where s is the dimension of the vector space V defined in Section 2. As usual, we presuppose that, to factor a general polynomial using our method, the squarefree decomposition is obtained first. For information about square-free decomposition, we refer the reader to [7], and the references therein. Moreover, we may

Computational experiments

In this section, we present the results of some computational experiments comparing the time required by Maple to both run Algorithm 1 and then factor each of the polynomials in the output with the time elapsed as Maple calculated this factorization directly. These experiments are meant to be a simple illustration and therefore deal with relatively small polynomials. Experiments were run using Maple 15 in a personal computer.2

Final remarks

In this paper, we introduced a pre-test for bivariate polynomial factorization over F2, which combines the basic structure of an algorithm due to Lecerf [14] with ideas of Gao [5]. It is not a coincidental that the factorization obtained by Algorithm 1 has been presented in connection with the merge-free factorization induced by the corresponding substitution. We believe that these two factorizations coincide, as computational evidence suggests that Theorem 1 may be strengthened as follows.

Conjecture 7

Let

References (18)

There are more references available in the full text version of this article.

Cited by (0)

1

C. Hoppen was partially supported by FAPERGS (Proc. 2233-2551/ 14-9) and CNPq (Proc. 448754/2014-2 and 308539/2015-0).

View full text