Factorization Of Polynomials Research Articles

Efficient decomposition of separable algebras

We present new, efficient algorithms for computations on separable matrix algebras over infinite fields. We provide a probabilistic method of the Monte Carlo type to find a generator for the center of a given algebra A⊆ F m×m over an infinite field F . The number of operations used is within a logarithmic factor of the cost of solving m× m systems of linear equations. A Las Vegas algorithm is also provided under the assumption that a basis and set of generators for the given algebra are available. These new techniques yield a partial factorization of the minimal polynomial of the generator that is computed, which may reduce the cost of computing simple components of the algebra in some cases.

On the span in channel assignment problems: bounds, computing and counting

The channel assignment problem involves assigning radio channels to transmitters, using a small span of channels but without causing excessive interference. We consider a standard model for channel assignment, the constraint matrix model, which extends ideas of graph colouring. Given a graph G=( V, E) and a length l( uv) for each edge uv of G, we call an assignment φ: V→{1,…, t} feasible if | φ( u)− φ( v)|⩾ l( uv) for each edge uv. The least t for which there is a feasible assignment is the span of the problem. We first derive two bounds on the span, an upper bound (from a sequential assignment method) and a lower bound. We then see that an extension of the Gallai-Roy theorem on chromatic number and orientations shows that the span can be calculated in O( n!) steps for a graph with n nodes, neglecting a polynomial factor. We prove that, if the edge-lengths are bounded, then we may calculate the span in exponential time, that is, in time O( c n ) for a constant c. Finally we consider counting feasible assignments and related quantities.

On hopf bifurcations in singularly perturbed systems

It has been shown recently that, under some generic assumptions, there exists a Hopf curve /spl lambda/ = /spl lambda/ (/spl epsiv/) for singularly perturbed systems of the form x/spl dot/ = f (x, y, /spl lambda/), /spl epsiv/y/spl dot/ = g(x, y, /spl lambda/) near the singular surface defined by det g/sub v/ = 0. In this note, we are concerned with the Hopf curve and obtain three results: 1) we prove that the eigenvalue crossing condition for the Hopf curve holds without additional assumption; 2) we provide an improved form of an existing derivative formula for the Hopf curve which is more suitable for practical computations; and 3) we give a quite precise description of the spectrum structure of the linearization along the Hopf curve. All three results (stated in the main theorem) are useful for a better understanding of Hopf bifurcations in singularly perturbed systems. Our analysis is based on a factorization of parameter dependent polynomials (Lemma 2.3).

Factoring multivariate polynomials via partial differential equations

A new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic. It is based on a simple partial differential equati...

Polynomial factorization through Toeplitz matrix computations

The problem of polynomial factorization is translated into the problem of constructing a Wiener–Hopf factorization, and three algorithms are designed for the solution of the latter problem. These algorithms are based on solving linear systems with large (but finite) circulant and Toeplitz matrices. The algorithms are of low complexity and, perhaps most importantly, they are extremely lucid. An upper bound for the condition number of the problem of polynomial factorization is given in terms of the condition number of a certain Toeplitz matrix.

Homogeneous Polynomials and the Minimal Polynomial of COS $(2\pi / n)$

It is well known that if $\Phi _n (x)$ is the $n$th cyclotomic polynomial, then there is a factorization $x^n - 1 = \prod \Phi _d (x)$, where the product is taken over the divisors $d$ of $n$. Thus, one can obtain, by Möbius inversion, a product formula for each $\Phi _n (x)$ in terms of the various factors $x^d - 1$. The purpose of this note is two-fold. First, we show that the above factorization implies a similar factorization for the minimal polynomials of the algebraic numbers $\cos (2\pi / n)$, where $n$ is a positive integer. Secondly, we give an explicit formula for the minimal polynomials of $\cos (2\pi / p)$, where $p$ is prime.

A genetic algorithm approach to the problem of factorization of general multidimensional polynomials

In this paper, a solution to the problem of the multidimensional (m-D) polynomial factorization is attempted by using genetic algorithms (GAs). The proposed method is based on an appropriate minimization of the norm of the difference between the original polynomial and its desirable factorized form. Using GAs, we can obtain better results than with other methods of minimization (numerical techniques, neural networks, etc.). The present methodology, which can also be used for every type of m-D factorization, is illustrated by means of a numerical example.

Equalities among Capacities of (d,k)-Constrained Systems

In this paper, we consider the problem of determining when the capacities of distinct (d,k)-constrained systems can be equal. A (d,k)-constrained system consists of binary sequences which have at least d zeros and at most k zeros between any two successive ones. If we let C(d,k) denote the capacity of a (d,k)-constrained system, then it is known that C(d,2d) = C(d+1,3d+1) and C(d,2d+1) = C(d+1,\infty)$. Repeated application of these two identities also yields the chain of equalities C(1,2) = C(2,4) = C(3,7) = C(4,\infty)$. We show that these are the only equalities possible among the capacities of (d,k)-constrained systems. In the process, we also provide useful factorizations of the characteristic polynomials for these constraints.

Polynomial Interpolation on the Unit Sphere

The problem of interpolation at (n+1)2 points on the unit sphere S2 by spherical polynomials of degree at most n is studied. Many sets of points that admit unique interpolation are given explicitly. The proof is based on a method of factorization of polynomials. A related problem of interpolation by trigonometric polynomials is also solved.

Normal factorisation of polynomials and computational issues

This paper introduces the notion of normal factorisation of polynomials and then presents a new numerical algorithm for the computation of this factorisation. This procedure avoids computation of roots of polynomials and it is based on the use of algorithms determining the greatest common divisor of polynomials. In this paper, the general aspects of the algorithm are discussed and a symbolic implementation is given. The theoretical algorithm provides the basis for a numerical procedure, the general aspects of which are also discussed here. 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 then be 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 an integral part of computations for algebraic control problems.

Worst-case time bounds for coloring and satisfiability problems

We consider worst case time bounds for certain NP-complete problems. In particular, we consider the ( k,2)-satisfiability problem which includes as special cases some canonical problems such as graph coloring and satisfiability. For the ( k,2)-satisfiability problem, we present a randomized algorithm that runs in time O ∗((k!) n/k) . 2 2 Throughout this paper, O ∗(f(n)) will denote O( n c f( n)). We also note that the number of clauses m is polynomially bounded in n and k, and hence can be ignored when polynomial factors are ignored. This bound is equivalent to O(( k/ c k ) n ) with c k increasing to the asymptotic limit e. For k⩾11, we improve upon the O((0.4518 k) n ) randomized bound of Eppstein [Proceedings of the 12th Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 329–337]. A special case of ( k,2)-satisfiability is k-colorability; here we achieve the above time bound for a slightly larger c k that has the same asymptotic behavior.

Prime-length real-valued polynomial residue division algorithms

One class of efficient algorithms for computing a discrete Fourier transform (DFT) is based on a recursive polynomial factorization of the polynomial 1-z/sup -N/. The Bruun algorithm is a typical example of such algorithms. Previously, the Bruun algorithm, which is applicable only when system lengths are powers of two in its original form, is generalized and modified to be applicable to the case when the length is other than a power of two. This generalized algorithm consists of transforms T/sub d,f/ with prime d and real f in the range 0/spl les/f<0.5. T/sub d,0/ computes residues X(z)mod(1-z/sup -2/) and X(z)mod(1-2 cos(/spl pi/k/d)z/sup -1/+z/sup -2/), k=1, 2, ..., d-1, and T/sub d,f/ (f /spl ne/0) computes residues X(z)mod(1-2cos(2/spl pi/(f+k)/d)z/sup -1/+z/sup -2/), k=0, 1, ..., d-1 for a given real signal X(z) of length 2d. The purpose of this paper is to find efficient algorithms for T/sub d,f/. First, polynomial factorization algorithms are derived for T/sub d,0/ and T/sub d,1/4/. When f is neither 0 nor 1/4, it is not feasible to derive a polynomial factorization algorithm. Two different implementations of T/sub d,f/ for such f are derived. One implementation realizes T/sub d,f/ via a d-point DFT, for which a variety of fast algorithms exist. The other implementation realizes T/sub d,f/ via T/sub d, 1/4/, for which the polynomial factorization algorithm exists. Comparisons show that for d/spl ges/5, these implementations achieve better performance than computing each output of T/sub d,f/ separately.

Computing the Primary Decomposition of Zero-dimensional Ideals

Let K be an infinite perfect computable field and let I⊆K [ x ] be a zero-dimensional ideal represented by a Gröbner basis. We derive a new algorithm for computing the reduced primary decomposition of I using only standard linear algebra and univariate polynomial factorization techniques. In practice, the algorithm generally works in finite fields of large characteristic as well.

A deterministic (2−2/( k+1)) n algorithm for k-SAT based on local search

Local search is widely used for solving the propositional satisfiability problem. Papadimitriou (1991) showed that randomized local search solves 2-SAT in polynomial time. Recently, Schöning (1999) proved that a close algorithm for k-SAT takes time (2−2/ k) n up to a polynomial factor. This is the best known worst-case upper bound for randomized 3-SAT algorithms (cf. also recent preprint by Schuler et al.). We describe a deterministic local search algorithm for k-SAT running in time (2−2/( k+1)) n up to a polynomial factor. The key point of our algorithm is the use of covering codes instead of random choice of initial assignments. Compared to other “weakly exponential” algorithms, our algorithm is technically quite simple. We also describe an improved version of local search. For 3-SAT the improved algorithm runs in time 1.481 n up to a polynomial factor. Our bounds are better than all previous bounds for deterministic k-SAT algorithms.

The Calculation of Radical Ideals in Positive Characteristic

We propose an algorithm for computing the radical of a polynomial ideal in positive characteristic. The algorithm does not involve polynomial factorization.

Interval Arithmetic in Cylindrical Algebraic Decomposition

Cylindrical algebraic decomposition requires many very time consuming operations, including resultant computation, polynomial factorization, algebraic polynomial gcd computation and polynomial real root isolation. We show how the time for algebraic polynomial real root isolation can be greatly reduced by using interval arithmetic instead of exact computation. This substantially reduces the overall time for cylindrical algebraic decomposition.

Sobre los números de Hal y Lah

En este trabajo se estudian algunas propiedades relativas a los números de Hal (Hkn ) y de Lah (Lkn), que son aquellos números que surgen en combinatoria asociados con los polinomios factoriales hacia arriba y hacia abajo.&#x0D; Se establecen las fórmulas por recurrencia, las funciones generadoras y las fórmulas cerradas para esta familia de n´úmeros , entre otras propiedades.

Synthesis of robust strictly positive real discrete-time systems with l/sub 2/ parametric perturbations

The author addresses the robust strict-positive realness problem for families of discrete-time polynomials where the uncertainty is described via a l/sub 2/ ball in coefficient space. It is shown constructively that under the only assumption that all the polynomials of the family are Schur, the sought filter can be provided in closed form as a polynomial or rational function with an a-priori bounded degree. The proposed synthesis procedure is based on the solution of a simple polynomial factorization problem.

Factorization of monic polynomials

We prove a uniqueness result about the factorization of a monic polynomial over a general commutative ring into comaximal factors. We apply this result to address several questions raised by Steve McAdam. These questions, inspired by Hensel's Lemma, concern properties of prime ideals and the factoring of monic polynomials modulo prime ideals.

Polynomial factorization over ${\mathbb F}_2$

We describe algorithms for polynomial factorization over the binary field F2, and their implementation. They allow polynomials of degree up to 250 000 to be factored in about one day of CPU time, distributing the work on two processors.