Bounds on numers of vectors of multiplicities for polynomials which are easy to compute (original) (raw)

On factoring parametric multivariate polynomials

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.

A note on Gao’s algorithm for polynomial factorization

Theoretical Computer Science, 2011

Shuhong Gao (2003) [6] has proposed an efficient algorithm to factor a bivariate polynomial f over a field F. This algorithm is based on a simple partial differential equation and depends on a crucial fact: the dimension of the polynomial solution space G associated with this differential equation is equal to the number r of absolutely irreducible factors of f. However, this holds only when the characteristic of F is either zero or sufficiently large in terms of the degree of f. In this paper we characterize a vector subspace of G for which the dimension is r, regardless of the characteristic of F, and the properties of Gao's construction hold. Moreover, we identify a second vector subspace of G that leads to an analogous theory for the rational factorization of f .

Complexity bound for the absolute factorization of parametric polynomials

Journal of Mathematical Sciences, 2006

An algorithm is constructed for the absolute factorization of polynomials with algebraically independent parametric coefficients. It divides the parameter space into pairwise disjoint pieces such that the absolute factorization of polynomials with coefficients in each piece is given uniformly. Namely, for each piece there exist a positive integer l ≤ d, l variables C1,. .. , Cl algebraically independent over the ground field F , and rational functions bJ,j of the parameters and of the variables C1,. .. , Cl such that for any parametric polynomial f with coefficients in this piece, there exist c1,. .. , cl ∈ F with f = j Gj , where Gj = |J| BJ,j Z J is absolutely irreducible. Here Z = (Z0,. .. , Zn) are the variables of f , each BJ,j is the value of bJ,j at the coefficients of f and c1,. .. , cl, and F denotes the algebraic closure of F. The number of pieces does not exceed (2d 2 +1) 2n+3d+5 , and the algorithm performs d O(ndr 2) arithmetic operations in F (thus the number of operations is exponential in the number r = n+1+d n+1 of coefficients of f), and its binary complexity is bounded by d O(ndr 2) if F = Q and by pd ndr 2 O(1) if F = Fp, where d is an upper bound on the degrees of polynomials. The techniques used include the Hensel lemma and the quantifier elimination in the theory of algebraically closed fields. Bibliography: 20 titles.

On factorization of multivariate polynomials over algebraic number and function fields

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.

On the complexity of the multivariate resultant

Journal of Complexity, 2013

The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n homogeneous equations in n variables is satisfiable (the resultant is a polynomial in the system's coefficients which vanishes if and only if the system is satisfiable). In this paper, we investigate the complexity of computing the multivariate resultant.

Complexity of solving parametric polynomial systems

We present three algorithms in this paper: the first algorithm solves zero-dimensional parametric homogeneous polynomial systems with single exponential time in the number n of the unknowns, it decomposes the parameters space into a finite number of constructible sets and computes the finite number of solutions by parametric rational representations uniformly in each constructible set. The second algorithm factorizes absolutely multivariate parametric polynomials with single exponential time in n and in the degree upper bound d of the factorized polynomials. The third algorithm decomposes the algebraic varieties defined by parametric polynomial systems of positive dimension into absolutely irreducible components uniformly on the values of the parameters. The complexity bound of this algorithm is double-exponential in n. On the other hand, the complexity lower bound of the problem of resolution of parametric polynomial systems is double-exponential in n.

On multiplicities in polynomial system solving

Transactions of the American Mathematical Society, 1996

This paper deals with the description of the solutions of zero dimensional systems of polynomial equations. Based on different models for describing solutions, we consider suitable representations of a multiple root, or more precisely suitable descriptions of the primary component of the system at a root. We analyse the complexity of finding the representations and of algorithms which perform transformations between the different representations.

A numerical-symbolic algorithm for computing the multiplicity of a component of an algebraic set

Journal of Complexity, 2006

Let F1, F2, . . . , Ft be multivariate polynomials (with complex coefficients) in the variables z 1 , z 2 , . . . , z n . The common zero locus of these polynomials, V (F 1 , F 2 , . . . , F t ) = {p ∈ C n |F i (p) = 0 for 1 ≤ i ≤ t}, determines an algebraic set. This algebraic set decomposes into a union of simpler, irreducible components. The set of polynomials imposes on each component a positive integer known as the multiplicity of the component. Multiplicity plays an important role in many practical applications. It determines roughly "how many times the component should be counted in a computation." Unfortunately, many numerical methods have difficulty in problems where the multiplicity of a component is greater than one. The main goal of this paper is to present an algorithm for determining the multiplicity of a component of an algebraic set. The method sidesteps the numerical stability issues which have obstructed other approaches by incorporating a combined numerical-symbolic technique.

On square-free factorization of multivariate polynomials over a finite 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.

A Survey on the Complexity of Solving Algebraic Systems

International Mathematical Forum, 2010

This paper presents a lecture on existing algorithms for solving polynomial systems with their complexity analysis from our experiments on the subject. It is based on our studies of the complexity of solving parametric polynomial systems. It is intended to be useful to two groups of people: those who wish to know what work has been done and those who would like to do work in the field. It contains an extensive bibliography to assist readers in exploring the field in more depth. The paper provides different methods and techniques used for representing solutions of algebraic systems that include Rational Univariate Representations (RUR), Gröbner bases, etc.