On factoring parametric multivariate polynomials

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.

ACM Sigsam Bulletin, 2009

This thesis presents an algorithm for factoring polynomials over the rationals which follows the approach of the van Hoeij algorithm. The key theoretical novelty in our approach is that it is set up in a way that will make it possible to prove a new complexity result for this algorithm which was actually observed on prior algorithms. One difference of this algorithm from prior algorithms is the practical improvement which we call early termination. Our algorithm should outperform prior algorithms in many common classes of polynomials (including irreducibles).

Japan Journal of Industrial and Applied Mathematics, 1991

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.

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.

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.

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, 1985

We consider the following problem : given a polynomial f(x) E k[x], k a field, find a complete decomposition off in the form f= g1 092 0 . . . o 91, where o denotes functional composition . After reviewing some known results about existence and uniqueness of such decompositions two algorithms are presented that solve the decomposition problem .

Lecture Notes in Computer Science, 2006

The Complete Root Classification for a univariate polynomial with symbolic coefficients is the collection of all the possible cases of its root classification, together with the conditions its coefficients should satisfy for each case. Here an algorithm is given for the automatic computation of the complete root classification of a polynomial with complex symbolic coefficients. The application of complete root classifications to some real quantifier elimination problems is also described.

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.

Elimination theory is at the origin of algebraic geometry in the 19-th century and deals with algorithmic solving of multivariate polynomial equation systems over the complex numbers, or, more generally, over an arbitrary algebraically closed field. In this paper we investigate the intrinsic sequential time complexity of universal elimination procedures for arbitrary continuous data structures encoding input and output objects of elimination theory (i.e. polynomial equation systems) and admitting the representation of certain limit objects. Our main result is the following: let be given such a data structure and together with this data structure a universal elimination algorithm, say P , solving arbitrary parametric polynomial equation systems. Suppose that the algorithm P avoids "unnecessary" branchings and that P admits the efficient computation of certain natural limit objects (as e.g. the Zariski closure of a given constructible algebraic set or the parametric greatest common divisor of two given algebraic families of univariate polynomials). Then P cannot be a polynomial time algorithm. The paper contains different variants of this result which are formulated and discussed both from the point of view of exact (i.e. symbolic) as well as from the point of view of approximative (i.e. numeric) computing. The mentioned results shall only be discussed informally. Proofs will appear elsewhere.

Journal of Symbolic Computation, 2007

We present a new algorithm for solving basic parametric constructible or semi-algebraic

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.

IFAC Proceedings Volumes, 2001

The paper presents a numerical algorithm for the computation of the normal factorisation of polynomials. This procedure avoids computation of roots of polynomials and it is based on the use of algorithms determining the greatest common divisor of polynomials. More precisely, it applies algorithms suitable for the computation of the greatest common divisor of two polynomials that are based on resultant sets. The advantage of such a factorisation is that it handles the determination of multiplicities and produces factors of lower degree and with distinct roots. For such polynomials robust numerical techniques based on finding roots may be then used, if it is desired to work out the usual irreducible factorisation. A detailed description of the implementation of the algorithm is presented. The problem considered here is integral part of computations for algebraic control problems.