Showing papers on "Square-free polynomial published in 2000"

Mean value for the matching and dominating polynomial

Abstract: The mean value of the matching polynomial is computed in the family of all labeled graphs with n vertices. We introduce the dominating polynomial of a graph whose coefficients enumerate the dominating sets for a graph and study some properties of the polynomial. The mean value of this polynomial is determined in a certain special family of bipartite digraphs.

The permanental polynomial.

Abstract: This study identifies properties and uses of the permanental polynomial of adjacency matrixes of unweighted chemical graphs Coefficients and zeroes of the polynomial for several representative structures are provided, and their properties are discussed A computer program for calculating the permanental polynomial from the adjacency matrix is also described

On lattice reduction for polynomial matrices

Abstract: A simple algorithm for transformation to weak Popov form | essentially lattice reduction for polynomial matrices | is described and analyzed. The algorithm is adapted and applied to various tasks involving polynomial matrices: rank prole and determinant computation; unimodular triangular factorization; transformation to Hermite and Popov canonical form; rational and diophantine linear system solving; short vector computation.

Polynomial Eigenfunctions of the Calogero—Sutherland—Moser Models with Exchange Terms

Abstract: We examine eigenfunctions of the periodic and rational Calogero-Sutherland-Moser system with exchange terms. In particular explicit formulae for the normalization of Jack polynomials with prescribed symmetry (i.e., eigenfunctions that can be chosen to be symmetric or antisymmetric in certain variables) are given. In addition Macdonald polynomials of prescribed symmetry are considered, and it is shown that factorization can be achieved in certain cases.

Parallel Multiplication in GF(2^k) usingPolynomial Residue Arithmetic

Abstract: We present a novel method of parallelization of the multiplication operation in \GF(2^k) for an arbitrary value of k and arbitrary irreducible polynomial n(x) generating the field. The parallel algorithm is based on polynomial residue arithmetic, and requires that we find L pairwise relatively prime modulim_i(x) such that the degree of the product polynomialM(x)=m_1(x)m_2(x)\cdots m_L(x) is at least 2k. The parallel algorithm receives the residue representations of the input operands (elements of the field) and produces the result in its residue form, however, it is guaranteed that the degree of this polynomial is less than k and it is properly reduced by the generating polynomial n(x), i.e., it is an element of the field. In order to perform the reductions, we also describe a new table lookup based polynomial reduction method.

Encryption of programs represented as polynomial mappings and their computations

Abstract: Three variations of a method of representing (abstract) state machines as polynomial mappings, and three variations of a corresponding encryption program stored on a computer readable medium The encryption program is based directly on symbolic functional composition of polynomial mappings with permutations expressed as polynomial mappings

Approximate gcd of multivariate polynomials

Abstract: Given two polynomials F and G in R[x1, . . . , xn], we are going to find the nontrivial approximate GCD C and polynomials F , G ∈ R[x1, . . . , xn] such that ||F − CF ′|| 1. Approximate GCD computation of univariate polynomials provides the basis for solving multivariate problem. But it is nontrivial to modify the techniques used in symbolic computation such as interpolation and Hensel lifting to compute the approximate GCD of multivariate polynomials. In section 2, we briefly review two methods 2,10 for computing GCD of polynomials with floating-point coefficients. In section 3, we focus on extending the Hensel lifting technique to polynomials with floating-point coefficients. The method involves the QR decomposition of a Sylvester matrix. We propose an efficient new algorithm which exploits the special structure of Sylvester matrix. A local optimization problem is also been proposed to improve the candidate approximate factors obtained from Hensel lifting. In order to compare the performance of the different methods, we implement all three methods in Maple. In section 4, we summarize the special problems we encounter when they are applied to polynomials with floating-point coefficients. A set of examples are given to show the efficiency and stability of the algorithms.

Branch Points in One-Dimensional Gaussian Scale Space

Abstract: Scale space analysis combines global and local analysis in a single methodology by simplifying a signal. The simplification is indexed using a continuously varying parameter denoted scale. Different analyses can then be performed at their proper scale. We consider evolution of a polynomial by the parabolic partial differential heat equation. We first study a basis for the solution space, the heat polynomials, and subsequently the local geometry around a branch point in scale space. By a branch point of a polynomium we mean a scale and a location where two zeros of the polynomial merge. We prove that the number of branch points for a solution is \lfloor\frac{n}{2}\rfloor for an initial polynomial of degree n. Then we prove that the branch points uniquely determine a polynomial up to a constant factor. Algorithms are presented for conversion between the polynomial's coefficients and its branch points.

An extension of the Cobham-Semënov Theorem

Abstract: Let θ, θ′ be two multiplicatively independent Pisot numbers, and let U , U ′ be two linear numeration systems whose characteristic polynomial is the minimal polynomial of θ and θ′, respectively. For every n ≥ 1, if A ⊆ ℕ n is U -and U ′ -recognizable then A is definable in 〈ℕ: + 〉.

Finding edge-disjoint paths in partial k-trees

Abstract: For a given graph G and p pairs (s i ,t i ) , $1\leq i\leq p$ , of vertices in G , the edge-disjoint paths problem is to find p pairwise edge-disjoint paths P i , $1\leq i\leq p$ , connecting s i and t i . Many combinatorial problems can be efficiently solved for partial k -trees (graphs of treewidth bounded by a fixed integer k ), but the edge-disjoint paths problem is NP-complete even for partial 3 -trees. This paper gives two algorithms for the edge-disjoint paths problem on partial k -trees. The first one solves the problem for any partial k -tree G and runs in polynomial time if p=O( log n) and in linear time if p=O(1) , where n is the number of vertices in G . The second one solves the problem under some restriction on the location of terminal pairs even if $p\geq \log n$ .

Quadratic divisors of harmonic polynomials inR n

Abstract: Necessary and sufficient conditions are given for a quadratic polynomial to be a divisor of a nonzero harmonic polynomial inRn.

FFT-based fast polynomial rooting

Abstract: A fast, accurate and robust approach is proposed for the computation of the roots of complex polynomials. The method is derived from the DFT-based differential cepstrum estimation for moving average signals. This minimum/maximum-phase polynomial factorisation is easily extended to a factorisation along an arbitrary circle. In an iterative fashion, we estimate the largest root modulus from the differential cepstrum, then factor out the associated root(s) from the polynomial. For band-limited signals with roots located along the unit circle, the polynomial origin is slightly perturbed prior to the application of the algorithm. On average, three fast Fourier transforms are required per polynomial root, offering a significant computational advantage in the root estimation of moderate to high order polynomials.

Polynomial index growth groups

Abstract: We show that the order of a finite simple of Lie type is bounded by a small constant power of its exponent. This confirms, in a strengthened form, a conjecture of Vaughan-Lee and Zel'manov on the order and exponent of almost simple groups. We also obtain various structural restrictions on groups of polynomial index growth. Combining the above results we construct finitely generated residually finite groups of polynomial index growth which are neither linear nor boundedly generated. This answers questions of Segal and Platonov–Rapinchuk respectively. A further question of Platonov–Rapinchuk concerning a weakened polynomial index growth assumption is also answered.

A constructive algorithm for 2-D spectral factorization with rational spectral factors

Abstract: Spectral factorization of para-Hermitian polynomial matrices nonnegative on the imaginary axis is known to play a crucial role in signal and system theory. Such factorization, although always feasible in one-dimensional (1-D) case, can be carried out in two-dimensional (2-D) case in a somewhat modified form. In this modified form, even spectral factors of scalar polynomials must be rational matrices (with normal rank one), analytic in the Cartesian products of open right half planes. Despite this constrained form, such spectral factors are important in the context of 2-D systems. The feasibility of such spectral factorization is apparent in view of classical results from algebraic geometry, but a constructive proof of it has not been available in system theoretic literature. By a selection of elementary techniques borrowed from different sources, we give a constructive algorithm for the aforementioned spectral factorization of 2-D para-Hermitian scalar positive definite polynomials, and thus, indirectly provide a constructive proof of the corresponding result for para Hermitian polynomial matrices as well. An example illustrating the main aspect of the algorithm is included. Analog of the result for discrete time systems, and comparisons with known 2-D spectral factorization results of other types are also included.

Dynamics on Hubbard trees

Abstract: It is well known that the Hubbard tree of a postcritically finite complex polynomial contains all the combinatorial information on the polynomial In fact, an abstract Hubbard tree as defined in (23) uniquely determines the polynomial up to affine conjugation In this paper we give necessary and sufficient conditions enabling one to deduce directly from the restriction of a quadratic Misiurewicz polynomial to its Hubbard tree whether the polynomial is renormalizable, and in this case, of which type Moreover, we study dynamical features such as entropy, transitivity or periodic structure of the polynomial restricted to the Hubbard tree, and compare them with the properties of the polynomial on its Julia set In other words, we want to study how much of the "dynamical information" about the polynomial is captured by the Hubbard tree 1 Introduction In this paper we deal with Hubbard trees of Mi- siurewicz polynomials, for the most part of degree two, and the dynamical properties of such a polynomial f when restricted to its Hubbard tree H = H(f) A Hubbard tree and the restricted map catch the essence of the dynamics of f Indeed, from the combinatorial information given by a map on an abstract Hubbard tree (satisfying certain conditions) one can obtain the affine class of the actual polynomial realizing the tree asits Hubbard tree But since it is easier to deal with the dynamics on the tree, it is of interest to describe how one reads off properties of the polynomial f directly from the dynamics of the tree map f |H(f) The main results of the paper (see Subsection 15) give necessary and sufficient conditions enabling one to deduce directly fromf |H(f) whether the polynomial is renormalizable, and of which type Renormalization is a very important concept in holomorphic dynamics; it is therefore of interest to have a purely combinatorial characterization of this notion Our results also show that other dynamical properties of the polynomial on its Julia set, such as density of periodic points, total transitivity or maximal topological

On P Systems with Active Membranes Solving the Integer Factorization Problem in a Polynomial Time

Abstract: There are presented deterministic P systems with active membranes which are able to solve the Integer Factorization Problem in a polynomial time, which is the main result of the paper. There is introduced a class of programs for the description and correct implementation of algorithms of elementary number theory in nonstandard computing systems, especially in P systems with active membranes. By using some of these programs there is achieved the main result.

Polynomial inverse integrating factors

Abstract: Let (P, Q) be a C1 vector field defined in an open subset U IR2 We call inverse integrating factor a C1 solution V(x, y) of the equation In previous works it has been shown that this function plays an important role in the problem of the center and in the determination of limit cycles In this paper we obtain necessary conditions for a polynomial vector field (P, Q) to have a polynomial inverse integrating factor

On four-sheeted polynomial mappings of $ \mathbb C^2$. II. The general case

Abstract: In this paper we prove that there are no four-sheeted polynomial mappings of into itself whose Jacobian is a non-zero constant.

Submodules of the Hardy space over polynomial algebras

Abstract: The celebrated theorem of Beurling [3] states that any closed z-invariant subspace of the Hardy space H2 in the unit disk is of the form M = g ·H2 where g is a classical inner function. The result of Apostol, Bercovici, Foias, and Pearcy [2] emphasized the importance of a Beurling type theorem for the Bergman space A2 in the unit disk, and such a theorem was proved in 1996 by by Aleman, Richter, and Sundberg [1]. Their approach uses the concept of wandering property introduced by Halmos in 1961 [7]. More precisely the result in [1] states that if M is a closed z-invariant subspace of A2, then the set M zM generates M in the sense that M is the minimal closed z-invariant subspace of A2 which contains M zM . This gives rise to the following interpretation of a Beurling type theorem. Let C[z] stand, as usual, for the ring of polynomials. Then both the Hardy and Bergman spaces have a natural C[z]-module structure. Any z-invariant subspace corresponds to a C[z]-submodule in this setting. Thus a Beurling type theorem describes a constructive way of obtaining a generating set of closed submodules. In the Hardy setting for a closed C[z]-submodule M = g ·H2, M zM has dimension 1 and is spanned by g. This leads to the following general question. Let A be a subalgebra of H∞; then both Hardy and Bergman spaces can be considered as modules over A. Then given a closed A-submoduleM of the Hardy (Bergman) space, how can one describe a canonical procedure of finding a set of generators of M? In particular, is every closed A-submodule finitely generated, and if A0 = {f ∈ A : f(0) = 0}, must M A0M generate M as an A-submodule? We single out zero to follow the classical route for a canonical construction, but we could replace it with any point w in the unit disk, and all the results of this paper would remain valid.

Arbitrary polynomial transformations

Abstract: Previously, we looked at several polynomial transformations-the classical lowpass-to-highpass, lowpass-to-bandpass, and lowpass-to-bandstop transformations, and the special case of the bilinear transformation. We presented specific algorithms for accomplishing the transformations numerically. In this article, we end our preoccupation with polynomial transformations by presenting a straightforward algorithm derived from Heinen and Siddique [1988] for performing arbitrary polynomial transformations numerically, and, which is, in fact, based on Waggener's method [1980]. However, we do not implement Heinen and Siddique's algorithm directly. Instead, we rely on functions written to find sum-polynomials and product-polynomials. Our approach is quite as efficient, but it should prove easier to follow and code.

On nD polynomial matrix factorizations

Abstract: This paper presents an overview of one of the nD polynomial matrix factorization problems, namely the factorization of a given nD polynomial matrix A(z) into A(z)=A/sub 1/(z)A/sub 2/(z) such that A/sub 1/(z) and A/sub 2/(z) are both nD polynomial matrices with certain special properties. The paper covers applications of both "classical" techniques as well as the Grobner bases to this factorization problem for the 2D and nD (n>2) cases. The difference between the 2D and the nD cases is pointed out. Besides reviewing the existing results, a new result and some open problems are also presented.

Simple computational methods for polynomial interpolations

Abstract: Polynomial interpolation is the general approach to many control designs. Designs using the root locus, for example, directly apply polynomial interpolations. It is shown that such interpolations can be easily calculated using polynomial root finders. While this may appear obvious, it has previously not been approached this way. These results may be used in computer-aided control-system design software.