Modular Las Vegas algorithms for polynomial absolute factorization

数式処理, 2001

This article tests empirically two "dirty tricks" for the trial-division step of Berlekamp-Hensel type algorithm for the univariate polynomial factorization over Z. The tricks are 1) divisibility check of the constant term and 2) boundedness check of the second coefficient. So far, it has been said that 1) is quite effective but 2) is not so effective. However, defining the upper bound of the second coefficient suitably, we show by many examples that the trick 2) is also quite effective for polynomials of medium and large degrees, such as degree ≥ 15.

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.

2011

Any non constant polynomial over a field can be expressed as a product of ir-reducible polynomials. In finite fields, some algorithms work for the calculation of irreducible factors of a polynomial of positive degree. The factorization of polynomials over finite fields has great ...

Theoretical Computer Science, 1997

In this paper we present a new deterministic algorithm for computing the square-free decomposition of multivariate polynomials with coefficients from a finite field. Our algorithm is based on Yun's square-free factorization algorithm for characteristic 0. The new algorithm is more efficient than existing, deterministic algorithms based on Musser's squarefree algorithm. We will show that the modular approach presented by Yun has no significant performance advantage over our algorithm. The new algorithm is also simpler to implement and it can rely on any existing GCD algorithm without having to worry about choosing "good" evaluation points. To demonstrate this, we present some timings using implementations in Maple (Char et al., 1991), where the new algorithm is used for Release 4 onwards, and Axiom (Jenks and Sutor, 1992) which is the only system known to the author to use an implementation of Yun's modular algorithm mentioned above.

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 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 .

Mathematics of Computation, 1985

We present a probabilistic algorithm that finds the irreducible factors of a bivariate polynomial with coefficients from a finite field in time polynomial in the input size, i.e., in the degree of the polynomial and log (cardinality of field). The algorithm generalizes to multivariate polynomials and has polynomial running time for densely encoded inputs. A deterministic version of the algorithm is also discussed, whose running time is polynomial in the degree of the input polynomial and the size of the field.

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.

Journal of Number Theory, 2003

Ostrowski established in 1919 that an absolutely irreducible integral polynomial remains absolutely irreducible modulo all sufficiently large prime numbers. We obtain a new lower bound for the size of such primes in terms of the number of integral points in the Newton polytope of the polynomial, significantly improving previous estimates for sparse polynomials. where H(f) is the height of f , i.e. the maximum of the absolute values of its coefficients. In 1986, Ruppert [8] presented a sharper estimate: p > d 3d 2 −3 • H(f) d 2 −1 .

We study the problem of bounding a polynomial away from polynomials, which are absolutely irreducible. Such separation bounds are useful for testing whether a numerical polynomial is absolutely irreducible, given a certain tolarence on its coecients. Using the absolutely irreducible criterion due to Rupport, we are able to nd useful separation bounds, in several norms, for bivariate polynomials. We consider the problem of factoring a bivariate polynomial f(x; y) = Xm; n i=0; j=0 ci;jxiyj 2 C[x; y] where the actual coecients of f are rational or complex numbers and the C is a eld of complex numbers. The idea is more precisely that we compute a B(f) 2 R > 0 such that all ~ f 2 C[x; y] with jjf 􀀀 ~ fjj < B(f); and deg( ~ f)  deg(f) must remain absolutely irreducible. If B(f) is not too small, one can then declare f as a numerically irreducible. The largest possible B(f) constitutes the distance to the nearest factorizable polynomial and can be called the radius of irreducibility.

Finite Fields and Applications, 2001

Results on the worst case behavior of the authors' extension (Theor. Comput. Sci. 234 (2000), 301-308) of Shoup's algorithm for factoring polynomials over finite prime fields (Inf. Process. Lett. 33 (1990), 261-267) are improved. Moreover, the consequences of the average case behavior of the extended algorithm for multivariate algorithms are described, and an extension of Lenstra's algorithm (Lond. Math. Soc, Lect. Note Ser. 154 (1990), 76-85) for root finding over finite prime fields is presented.

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.

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.

Communications in Computer and Information Science

Maple 2019 has a new multivariate polynomial factorization algorithm for factoring polynomials in Z[x 1 , x 2 , ..., x n ], that is, polynomials in n variables with integer coefficients. The new algorithm, which we call MTSHL, was developed by the authors at Simon Fraser University. The algorithm and its sub-algorithms have been published in a sequence of papers [3-5]. It was integrated into the Maple library in early 2018 by Baris Tuncer under a MITACS internship with Maplesoft. MTSHL is now the default factoring algorithm in Maple 2019. The multivariate factorization algorithm in all previous versions of Maple is based mainly on the work of Paul Wang in [6, 7]. Keith Geddes is the main author of the Maple code. The algorithm and sub-algorithms are described in Chapter 6 of [1]. Wang's algorithm is still available in Maple 2019 with the method="Wang" option to the factor command. Wang's method can be exponential in n the number of variables. MTSHL is a random polynomial time algorithm. In [3] we found that it is faster than previous polynomial time methods of Kaltofen [2] and Zippel [8] and competitive with Wang's method in cases where Wang's method is not exponential in n. Here we give an overview of the main idea in MTSHL. Let a be the input polynomial to be factored. Suppose a = f g for two irreducible factors f, g ∈ Z[x 1 ,. .. , x n ]. The multivariate polynomial factorization algorithm used in all computer algebra systems is based on Multivariate Hensel Lifting (MHL). For a description of MHL see Chapter 6 of [1]. MHL first chooses integers α 2 , α 3 ,. .. , α n that satisfy certain conditions and factors the univariate image a 1 = a(x 1 , α 2 ,. .. , α n) in Z[x 1 ]. Suppose a 1 (x 1) = f 1 (x 1)g 1 (x 1) and f 1 (x 1) = f (x 1 , α 2 ,. .. , α n) and g 1 (x 1) = g(x 1 , α 2 ,. .. , α n). Next MHL begins Hensel lifting. Wang's design of Hensel lifting recovers the variables x 2 ,. .. , x n in the factors f and g one at a time in a loop. Let us use the notation

We present a lattice algorithm specifically designed for some classical applications of lattice reduction. The applications are for lattice bases with a generalized knapsack-type structure, where the target vectors are boundably short. For such applications, the complexity of the algorithm improves traditional lattice reduction by replacing some dependence on the bit-length of the input vectors by some dependence on the bound for the output vectors. If the bit-length of the target vectors is unrelated to the bit-length of the input, then our algorithm is only linear in the bit-length of the input entries, which is an improvement over the quadratic complexity floating-point LLL algorithms. To illustrate the usefulness of this algorithm we show that a direct application to factoring univariate polynomials over the integers leads to the first complexity bound improvement since 1984. A second application is algebraic number reconstruction, where a new complexity bound is obtained as well.