Polynomial Factorization in Maple 2019

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...

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.

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.

Lecture Notes in Computer Science, 1997

We describe the Maple [23] implementation of multivariate factorization over general finite fields. Our first implementation is available in Maple V Release 3. We give selected details of the algorithms and show several ideas that were used to improve its efficiency. Most of the improvements presented here are incorporated in Maple V Release 4. In particular, we show that we needed a general tool for implementing computations in GF(pk)[xl, x2,..., x,,]. We also needed an efficient implementation of our algorithms in Zp[y][x] because any multivariate factorization may depend on several bivariate factorizations. The efficiency of our implementation is illustrated by the ability to factor bivariate polynomials with over a million monomials over a small prime field.

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.

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.

2020

Sparse multivariate Hensel lifting (SHL) algorithms are used in multivariate polynomial factorization as efficient randomized algorithms. They improve on Wang’s classical multivariate Hensel lifting which can be exponential in the number of variables for sparse factors.

Multidimensional Systems and Signal Processing, 1994

The problem of factorizing a multivariab!e or multidimensional (m-D) polynomial f (zl, z2 ..... Zm), with real or complex coefficients and independent variables, into a number of m-D polynomial factors that may involve any independent variable or combination of them is considered. The only restriction imposed is that all factors should be linear in one and the same variable (say Zl). This type of faetorization is very near to the most general type: N 1 [-ei, I elm f(z , ..... Zm) = H [ Z " i=1 e I =0 ern=O (e I ..... c m ) ~ (0 ..... 0) ai;el,e 2 ...... m fill "'" Zemm+ Ci I

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 of Symbolic Computation, 2020

The standard approach to factor a multivariate polynomial in Z[x1, x2,. .. , xn] is to factor a univariate image in Z[x1] then recover the multivariate factors from their univariate images using a process known as multivariate Hensel lifting. Wang's multivariate Hensel lifting recovers the variables one at a time. It is currently implemented in many computer algebra systems, including Maple, Magma and Singular. When the factors are sparse, Wang's approach can be exponential in the number of variables n. To address this, sparse Hensel lifting was introduced by Zippel and then improved by Kaltofen. Recently, Monagan & Tuncer introduced a new approach which uses sparse polynomial interpolation to solve the multivariate polynomial diophantine equations that arise inside Hensel lifting in random polynomial time. This approach is shown to be practical and faster than Zippel's and Kaltofen's algorithms and faster than Wang's algorithm for non-zero evaluation points. In this work we first present a complete description of the sparse interpolation used by Monagan & Tuncer and show that it runs in random polynomial time. Next we study what happens to the sparsity of multivariate polynomials when the variables are successively evaluated at numbers. We determine the expected number of remaining terms. We use this result to revisit and correct the complexity analysis of Zippel's original sparse interpolation. Next we present an average case complexity analysis of our approach. We have implemented our algorithm in Maple with some sub-algorithms implemented in C. We present some experimental data comparing our approach with Wang's method for both sparse and dense factors. The data shows that our method is always competitive with Wang's method and faster when Wang's method is exponential in n.

Japan Journal of Industrial and Applied Mathematics, 1991

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.

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 ...

2010

Let f (X, Y)∈ Z [X, Y] be an irreducible polynomial over Q. We give a Las Vegas absolute irreducibility test based on a property of the Newton polytope of f, or more precisely, of f modulo some prime integer p. The same idea of choosing ap satisfying some prescribed properties together with LLL is used to provide a new strategy for absolute factorization of f (X, Y). We present our approach in the bivariate case but the techniques extend to the multivariate case.

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.