Showing papers on "Square-free polynomial published in 1999"

Solving Zero-Dimensional Systems Through the Rational Univariate Representation

Abstract: This paper is devoted to the resolution of zero-dimensional systems in K[X 1, …X n ], where K is a field of characteristic zero (or strictly positive under some conditions). We follow the definition used in MMM95 and basically due to Kronecker for solving zero-dimensional systems: A system is solved if each root is represented in such way as to allow the performance of any arithmetical operations over the arithmetical expressions of its coordinates. We propose new definitions for solving zero-dimensional systems in this sense by introducing the Univariate Representation of their roots. We show by this way that the solutions of any zero-dimensional system of polynomials can be expressed through a special kind of univariate representation (Rational Univariate Representation): where (f,g,g 1, …,g n ) are polynomials of K[X 1, …, X n ]. A special feature of our Rational Univariate Representation is that we dont loose geometrical information contained in the initial system. Moreover we propose different efficient algorithms for the computation of the Rational Univariate Representation, and we make a comparison with standard known tools.

Shifted normal forms of polynomial matrices

Abstract: In t,his paper we st,ucly the problen~ of transforrniug, via invertible colu1tln opcrat.ious~ it matrix polyioruial into a varicty Of .shiftcd forms. Esarnplcs of forms c:overed in out frmwa-ork include a colunm rctluccd form: il triangular fornlz R I%!rInite IlOrInd fOrll1 or it Popov IlorIllal fOrIll alollg wit,11 their shifted courltcrpart,s. I3y obt.aiuiug tlcgrvc bounds for uuiniodiilar niiill,iplicrs of shifted Popor fornis we are able t,o c11lbct1 tlic probleni of conqmtiug il normal forni into 0Iic of deterniiJIing a sliift.ctl forni Of R InininIal pOl~IlOIIliill hiISiS fur all XWXiiltWl IIliLtris polyioruial. Shifted niiuind polynomial lXPX?S cm be conlpu1.d via sigma bases [2! 31 iIlld ill POpOv forui Vii1 Mahler s~sbenls [il. Tl ic d 1. t,t, cr Iut:t.lIotl gives a fractiorl-frw algorithm for computing niatris riormal forms.

Type Systems For Polynomial-time Computation

Abstract: This thesis introduces and studies a typed lambda calculus with higher-order primitive recursion over inductive datatypes which has the property that all definable number-theoretic functions are polynomial time computable. This is achieved by imposing type-theoretic restrictions on the way results of recursive calls can be used. The main technical result is the proof of the characteristic property of this system. It proceeds by exhibiting a category-theoretic model in which all morphisms are polynomial time computable by construction. The second more subtle goal of the thesis is to illustrate the usefulness of this semantic technique as a means for guiding the development of syntactic systems, in particular typed lambda calculi, and to study their meta-theoretic properties. Minor results are a type checking algorithm for the developed typed lambda calculus and the construction of combinatory algebras consisting of polynomial time algorithms in the style of the first Kleene algebra.

Algebraic Geometry and Computer Vision: Polynomial Systems, Real andComplex Roots

Abstract: We review the different techniques known for doing exact computations on polynomial systems. Some are based on the use of Grobner bases and linear algebra, others on the more classical resultants and its modern counterparts. Many theoretical examples of the use of these techniques are given. Furthermore, a full set of examples of applications in the domain of artificial vision, where many constraints boil down to polynomial systems, are presented. Emphasis is also put on very recent methods for determining the number of (isolated) real and complex roots of such systems.

Forests, colorings and acyclic orientations of the square lattice

Abstract: There is no known polynomial time algorithm which generates a random forest or counts forests or acyclic orientations in general graphs. On the other hand, there is no technical reason why such algorithms should not exist. These are key questions in the theory of approximately evaluating the Tutte polynomial which in turn contains several other specializations of interest to statistical physics, such as the Ising, Potts, and random cluster models.

Symbolic Recipes for Polynomial System Solving

Abstract: In many branches of science and engineering where mathematics is used, the resolution of a problem coming from practice is often reduced to the search of a solution for a system of (algebraic or differential) equations modelling the considered problem. From our point of view, to solve a polynomial system of equations is to rewrite it (i.e., to present it in a different form) in such a way that some ‘nontrivial’ information about its solutions can be derived from this new presentation. The information mentioned above can be related to the existence or non-existence of complex or real solutions, to the number of real or complex solutions, to the approximation of one or several solutions, etc.

Approximating SVP infty to within Almost-Polynomial Factors is NP-hard

Abstract: We show SVP∞ and CVP∞ to be NP-hard to approximate to within nc/log log n for some constant c > 0. We show a direct reduction from SAT to these problems, that combines ideas from [ABSS93] and from [DKRS99], along with some modifications. Our result is obtained without relying on the PCP characterization of NP, although some of our techniques are derived from the proof of the PCP characterization itself [DFK+99].

Methods and algorithms of solving spectral problems for polynomial and rational matrices

Abstract: In the present paper, methods and algorithms for numerical solution of spectral problems and some problems in algebra related to them for one- and two-parameter polynomial and rational matrices are considered. A survey of known methods of solving spectral problems for polynomial matrices that are based on the rank factorization of constant matrices, i.e., that apply the singular value decomposition (SVD) and the normalized decomposition (the QR factorization), is given. The approach to the construction of methods that makes use of rank factorization is extended to one- and two-parameter polynomial and rational matrices. Methods and algorithms for solving some parametric problems in algebra based on ideas of rank factorization are presented. Bibliography: 326titles.

Approximate polynomial decomposition

Abstract: where deg g < deg f , deg h < deg f , deg∆f ≤ deg f and ∆f is “small” with respect to the 2-norm of the vector of coefficients. In practice if ‖f‖ denotes the 2-norm of f , then we compute g and h such that ‖∆f‖ is a local minimum with respect to variations in g and h. This problem has been studied for exact polynomials and rational functions by several authors [1, 2, 8, 10, 15, 16]. There are several reasons why approximate polynomial decomposition interests us:

Numerical and symbolic computation of polynomial matrix determinant

Abstract: The determinant of a polynomial matrix is frequently computed in analysis and/or design of control systems via polynomial approach. The computation can either be done symbolically using general symbolic procedures for determinant (MATHEMATICA/sup TM/, MAPLE/sup TM/) or by special numeric procedures (POLYNOMIAL TOOLBOX FOR MATLAB/sup TM/). This paper aims to compare the performance of the symbolic procedure built-in Mathematica with the best existing numerical routine based on the Fast Fourier Transform algorithm (FFT), coded for this purpose also in Mathematica. The new tailored numerical algorithm appears to be substantially more efficient than the general-purpose symbolic one. As it is also reasonably accurate, it is recommended for industrial applications of polynomial matrices.

A factorization of the Conway polynomial

Abstract: It is shown that the Conway polynomial of a link is a product of two factors, the first of which is the Conway polynomial of a knot obtained by banding together the link components and the second is determined, via an explicit formula, by the \(\tilde\mu\)-invariants of the link. In particular we get a formula, in terms of the μ-invariants, for the first non-zero coefficient of the Conway polynomial. A similar formula is obtained for the multi-variable Alexander-polynomial.

The Supervisory Control Problem of Discrete Event Systems using Polynomial Methods

Abstract: This paper regroups various studies achieved around polynomial dynamical system theory. It presents the basic algebraic tools for the study of this particular class of discrete event systems. The polynomial dynamical systems are defined by polynomial equations over Z/3Z. Their study relies on concept borrowed from elementary algebraic geometry: varieties, ideals and morphisms. They are the basic tools that allow us to translate properties or specifications from a geometric description to suitable polynomial computations. In this paper, we more precisely describe the controller synthesis methodology. We specify the main requirements as simple properties, named control objectives, that the controlled plant has to satisfy.The plant is specified as a polynomial dynamical system over Z/3Z. The control of the plant is performed by restricting the controllable input values to values suitable with respect to the control objectives. This restriction is obtained by incorporating new algebraic equations into the initial polynomial dynamical system, which specifies the plant. Various kind of control objectives are considered, such as ensuring the invariance or the reachability of a given set of states, as well as partial order relation to be checked by the controlled plant.

On the order of vanishing at 1 of a polynomial

Abstract: Suppose thatP(z) is a polynomial of degreen with complex coefficients such that one of its extreme coefficients is maximal in absolute value. We prove that ifn is sufficiently large, then the order of vanishing of the polynomial at a point in the unit circle is less than\(21\sqrt n /13\).

Controllability of structured polynomial systems

Abstract: Two algorithms, based on the Grobner basis method, which facilitate the controllability analysis for a class of polynomial systems are presented. The authors combine these algorithms with some results on output dead-beat controllability in order to obtain sufficient, as well as necessary, conditions for complete and state dead-beat controllability for a surprisingly large class of polynomial systems. Our results are generically applicable to the class of polynomial systems in strict feedback form.

Factoring high-degree polynomials over f 2 with Niederreiter's algorithm on the IBM SP2

Abstract: A C implementation of Niederreiter's algorithm for factoring polynomials over F 2 is described. The most time-consuming part of this algorithm, which consists of setting up and solving a certain system of linear equations, is performed in parallel. Once a basis for the solution space is found, all irreducible factors of the polynomial can be extracted by suitable gcd-computations. For this purpose, asymptotically fast polynomial arithmetic algorithms are implemented. These include Karatsuba & Ofman multiplication, Cantor multiplication and Newton inversion. In addition, a new efficient version of the half-gcd algorithm is presented. Sequential run times for the polynomial arithmetic and parallel run times for the factorization are given. A new world record for polynomial factorization over the binary field is set by showing that a pseudo-randomly selected polynomial of degree 300000 can be factored in about 10 hours on 256 nodes of the IBM SP2 at the Cornell Theory Center.

Using minimal polynomial bases for fault diagnosis

Abstract: A fundamental part of a fault diagnosis system is the residual generator. Design of residual generators to achieve perfect decoupling in linear systems is considered. A new method, the minimal polynomial basis approach, is presented, where the residual generation problem is transformed into the problem of finding polynomial bases for null-spaces of polynomial matrices. This is a standard problem in established linear systems theory, which means that numerically efficient computational tools are generally available. It is shown that the minimal polynomial basis approach can find all possible residual generators, including those of minimal degree, and the solution has a minimal parameterization.

The location of the zeros of the higher order derivatives of a polynomial

Abstract: Let p(z) be a complex polynomial of degree n having k zeros in a disk D. We deal with the problem of finding the smallest concentric disk containing k 1 zeros of p(l)(z). We obtain some estimates on the radius of this disk in general as well as in the special case, where k zeros in D are isolated from the other zeros of p(z). We indicate an application to the root-finding algorithms.

On the Characteristic Polynomial of the j-TH Order Fibonacci Sequence

Abstract: The characteristic polynomial of the jth-order Fibonacci sequence or the Fibonacci j-polynomial, is defined to be $$F^j = x^j - x^{j -1} - x^{j-2} - \cdots x -1, \ \ j =2,3,\cdots$$ .

Solving Equivalence of Recurrent Sequences in Groups by Polynomial Manipulation

Abstract: Methods are described for reducing the equivalence problem for sequences defined by recurrence equations in various finitely generated groups to polynomial Grobner basis constructions. The methods are simple enough to allow a straigthforward implementation in computer algebra programs.

Localization of Roots of a Polynomial not Represented in Canonical Form.

Abstract: The root isolation problem for the polynomial equation not represented in the canonical form can sometimes be solved without evaluation of the coefficients of powers of the variable. We investigate the approach based on representing first the equation in the equivalent determinantal (Hankel or block Hankel) form, and employing then Hermite’s root separation method. We illustrate this for the problems of eigenvalues localization, estimation of sensitivity of the roots of the parameter dependent polynomial and nonlinear optimization.