Polynomial Factorization in Maple 2019

2008

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.

We comment on the implementation of various algorithms in multivariate polynomial theory. Specifically, we describe a modular computation of triangular sets and possible applications. Next we discuss an implementation of the F 4 algorithm for computing Gröbner bases. We also give examples of how to use Gröbner bases for vanishing ideals in polynomial and rational function interpolation.

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.

Proceedings of the 2009 international symposium on Symbolic and algebraic computation - ISSAC '09, 2009

We present an efficient algorithm for factoring a multivariate polynomial f ∈ L[x1,. .. , xv] where L is an algebraic function field with k ≥ 0 parameters t1,. .. , t k and r ≥ 0 field extensions. Our algorithm uses Hensel lifting and extends the EEZ algorithm of Wang which was designed for factorization over Q. We also give a multivariate p-adic lifting algorithm which uses sparse interpolation. This enables us to avoid using poor bounds on the size of the integer coefficients in the factorization of f when using Hensel lifting. We have implemented our algorithm in Maple 13. We provide timings demonstrating the efficiency of our algorithm.

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.

ACM Communications in Computer Algebra, 2011

We demonstrate new routines for sparse multivariate polynomial multiplication and division over the integers that we have integrated into Maple 14 through the expand and divide commands. These routines are currently the fastest available, and the multiplication routine is parallelized with superlinear speedup. The performance of Maple is significantly improved. We describe our polynomial data structure and compare it with Maple's. Then we present benchmarks comparing Maple 14 with Maple 13, Magma, Mathematica, Singular, Pari, and Trip.

ACM Communications in Computer Algebra, 2017

We employ two techniques to dramatically improve Maple's performance on the Fermat benchmarks for simplifying rational expressions. First, we factor expanded polynomials to ensure that gcds are identified and cancelled automatically. Second, we replace all expanded polynomials by new variables and normalize the result. To undo the substitutions, we use a C routine for sparse multivariate division by a set of polynomials. The resulting times for the first Fermat benchmark are a factor of 17x faster than Fermat and 39x faster than Magma.

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.

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.

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.

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

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.

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.