Showing papers on "Square-free polynomial published in 1993"

A state model for the multi-variable alexander polynomial

Abstract: We construct a vertex type state model in Turaev's sense for the multi-variable (non-reduced) Alexander polynomial. Our model is a colored version of the 6-vertex free fermion model. To show the correspondence of our model and the multi-variable Alexander polynomial, we introduce colored braid groups and their Magnus representations. By using this model, a new set of axioms for the multi-variable Alexander polynomial is obtained.

The Moment Problem for Polynomial Forms in Normal Random Variables

Abstract: Let $Y$ be a random variable defined by a polynomial $p(W)$ of degree $n$ in finitely many normally distributed variables. This paper studies which such variables $Y$ are "determinate," i.e., have probability laws uniquely determined by their moments. Extending results of Berg, which applied to powers of a single normal variable, we prove that (a) $Y$ is determinate if $n = 1, 2$ or if $n = 4$, with the essential support of the law of $Y$ strictly smaller than the real line, and (b) $Y$ is not determinate either if $n$ is odd $\geq 3$ or if $n$ is even $\geq 6$ such that $p(\mathbf{w})$ attains a finite minimum value. Some other polynomials $Y = p(\mathbf{W})$ with even degree $n \geq 4$ are proved not to be determinate.

A unified method for multivariate polynomial factorizations

Abstract: 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 approximate roots. In this paper, we show that various kinds of multivariate polynomial factorizations can be performed by this method. We present algorithms for factorization of multivariate polynomials over power-series rings, over the integers, over algebraic number fields including algebraically closed fields, and over algebraic function fields. Furthermore, we discuss applicability of this method to univariate polynomial factorization.

The number of irreducible factors of a polynomial. I

Abstract: Let F(x) be a polynomial with coefficients in an algebraic number field k. We estimate the number of irreducible cyclotomic factors of F in k[x], the number of irreducible noncyclotomic factors of F, the number of nth roots of unity among the roots of F, and the number of primitive nth roots of unity among the roots of F. All of these quantities are counted with multiplicity and estimated by expressions which depend explicitly on k, on the degree of F and height of F, and (when appropriate) on n. We show by constructing examples that some of our results are essentially sharp

Fast Parallel Computation of Characteristic Polynomials by Leverrier's POwer Sum Method Adapted to Fields of Finite Characteristic

Abstract: Symmetric polynomials over F p in n indeterminates x1,..., x n are expressible as rational functions of the first n power sums s j =x l j + ...+ x n j with exponents j not divisible by p. There exist fairly simple regular specializations of these power sums by elements from F p so that all denominators of such rational expressions remain nonzero. This leads to a new class of fast parallel algorithms for the determinant or the characteristic polynomial of n×n matrices over any field of characteristic p with arithmetic circuit depth θ(logn)2.

Piecewise polynomial functions

Abstract: A continuous function f:ℝn→ℝ is piecewise polynomial if there are a finite semi-algebraic cover ℝ = M1U...UMr and polynomials f1,...,fr ∈ ℝ[X1,...,Xn] such that f ∣Mi = fi∣Mi for every i=1,...,r. Virtually the same definition can be used to explain the notion of a piecewise polynomial function on an affine semi-algebraic space over an arbitrary real closed field (cf. [6]. But there is a much more sweeping generalization if one uses the real closure R(A) (also called the ring of abstract semi-algebraic functions) associated with a ring A (cf. [3], [5], [21]). The definition of R(A) depends on the real spectrum Sper(A) of A (cf. [2], [4]). The real spectrum is a functor from rings to spectral spaces ([12]). For a ring A the points of Sper(A) may be thought of as prime ideals pCA together with a total order on the residue field qf(A/p). If α denotes a point of Sper(A) then the prime ideal is called the support of α, denoted by supp(α), and the real closure of qf(A/supp(α)) with respect to the total order specified by α is denoted by p(α). By definition, R(A) is a subring of the direct product of all the fields p(α) ([21, Chapter I). Although this construction is defined for arbitrary commutative rings it is of significance only if the real spectrum is not empty.

Computer Algebra, Symmetry Analysis and Integrability of Nonlinear Evolution Equations

Abstract: A computer algebra-aided symmetry approach to investigating integrability of polynomial-nonlinear evolution equations in one-temporal and one-spatial dimensions is presented. The approach is based on verifying the existence of higher conservation laws and symmetries. If the equations contain arbitrary numerical parameters, the problem of selection of all the integrable cases is reduced to the solving polynomial equations in those parameters. The Grobner basis technique is used in order to simplify and to solve such polynomial systems which typically have infinitely many solutions.

On primitive factorizations for n-D polynomial matrices

Abstract: In the study of the analysis, synthesis and realization of multivariate networks, n-dimensional (n-D) systems stability theory and feedback control, and n-D signal processing, it is often necessary to consider the feasibility of a factorization for a given n-D polynomial matrix A(z) in the form A/sub 1/(z) A/sub 2/(z), where A/sub 1/(z) and AA/sub 2/(z) are n-D polynomial matrices. The author reports on one such factorizations, i.e., the primitive factorization. A criterion for the existence of primitive factorizations for a class of n-D polynomial matrices is presented. The criterion can be used to construct a primitive factorization, when it exists, for an n-D polynomial matrix in this class. >

Factorization in Fq[x] and Brownian Motion

Abstract: We consider the set of polynomials of degree n over a finite field and put the uniform probability measure on this set. Any such polynomial factors uniquely into a product of its irreducible factors. To each polynomial we associate a step function on the interval [0,1] such that the size of each jump corresponds to the number of factors of a certain degree in the factorization of the random polynomial. We normalize these random functions and show that the resulting random process converges weakly to Brownian motion as n → ∞. This result complements earlier work by the author on the order statistics of the degree sequence of the factors of a random polynomial.

Polynomial LQG regulator design for general system configurations

Abstract: A polynomial-equation approach to the linear quadratic Gaussian (LQG) regulation problem for a general system configuration is presented. The solution is given in terms of a left-spectral factorization plus a pair of bilateral Diophantine equations. The resulting control-design procedure is based on an innovations representation of the system. This can be obtained from a physical description by solving, via polynomial equations, a minimum mean-square error filtering problem. The use in cascade of the above two procedures allows one to generalize previous polynomial design results to general system configurations. >

Explicitly integrable polynomial Hamiltonians and evaluation of Lie transformations.

Abstract: We have found that any homogeneous polynomial can be written as a sum of integrable polynomials of the same degree, with which each associated polynomial Hamiltonian is integrable, and the associated Lie transformation can be evaluated exactly. An integrable polynomial factorization has thus been developed to convert a sympletic map in the form of a Dragt-Finn factorization into a product of exactly evaluable Lie transformations associated with integrable polynomials. Having a small number of factorization bases of integrable polynomials enables one to consider a factorization with the use of high-order symplectic integrators so that a symplectic map can always be evaluated with the desired accuracy. The results are significant for studying the long-term stability of beams in accelerators.

Explicity integrable polynomial Hamiltonians and evaluation of Lie transformations

Abstract: We have found that any homogeneous polynomial can be written as a sum of integrable polynomials of the same degree, with which each associated polynomial Hamiltonian is integrable, and the associated Lie transformation can be evaluated exactly. An integrable polynomial factorization has thus been developed to convert a sympletic map in the form of a Dragt-Finn factorization into a product of exactly evaluable Lie transformations associated with integrable polynomials. Having a small number of factorization bases of integrable polynomials enables one to consider a factorization with the use of high-order symplectic integrators so that a symplectic map can always be evaluated with the desired accuracy

Optimization of the reliability polynomial in presence of mediocre elements

Abstract: Some connections between extremal set theory and the optimization of the reliability polynomial are shown. Then the concept of the reliability polynomial is generalized for the case when the elements can have three different states: good, mediocre and bad. The state of the device can be described by a 0,1,2 sequence. Such a state is called operative if the device operates when its elements are in the states described by the sequence. The maximum of the generalized reliability polynomial is studied under the condition that the set of operative states froms an antichain. © 1993 by John Wiley & Sons, Inc.

Polynomial Time Algorithm for the Equivalence of two Morphisms on Omega-Regular Languages

Abstract: Let L \(\subseteq\) Aω be an ω-regular language given by means of a non-deterministic Buchi automaton M. Let f,g: A∞ → B∞ be two morphisms. We give an algorithm to decide whether f and g are equivalent (word by word) on L. This algorithm has time complexity 0(mn3), where n is the number of arcs of M and m is the size of f and g. This result improves the only known algorithm for this problem which is exponential time [3].

On the uniqueness of the unital divisor of a matrix polynomial over an arbitrary field

Abstract: Let P be an arbitrary field and let P(x) be a ring of polynomials over P. Denote by Pn and Pn (x) the rings of n xn matrices over P and P(x), respectively. Consider a nonsingular polynomial matrix A(x) e P~ (x) (detA(x) $ 0) and represent it as a matrix polynomial over the field P: is a unital matrix polynomial. In this paper, we establish conditions under which a unital divisor D (x) extracted from the matrix polynomial A (x) is determined uniquely by its characteristic polynomial detA (x) = d(x). The re- sult obtained enables one to indicate the class of matrix polynomials for which this unital divisor is unique for a given characteristic polynomial. We also show how this result can be used when solving matrix polynomial equations. The uniqueness of a tmital divisor for a matrix polynomial over an algebraically closed field with char- acteristic zero was studied in ( 1 ). It is known (2) that, for a matrix A (x) e Pn (x), there exist matrices U(x), V(x) ~ GL n (Pn (x)) such that U (x)A (x) V(x) = F A (X) = diag (al(x), a2(x) ..... an(x)), where aj(x) e P(x), j= 1, 2 ..... n, are unital polynomials and ai(x ) (ai+l(x), i = 1, 2 ..... n - 1. The matrix FA(X ) is called a canonical diagonal form of the matrix A(x). It is also known that if a matrix is nonsingular, then it can be reduced to the normal Hermite form by right elementary transformations, i.e., for the matrix A (x), there exists a matrix W(x) e GLn (P(x)) such that

Roundoff-error reduction for evaluation of a function by polynomial approximation with error feedback in fixed-point arithmetic

Abstract: The relationship between Hornor's method for polynomial evaluation and a first-order recursive filter with error feedback (EFB) is described. It is shown that EFB is a useful technique for reducing the roundoff errors that occur in evaluating a function by polynomial approximation. >

Irregularities in the distribution of irreducible polynomials

Abstract: We prove that there exist monic polynomials f over GF(q) for which f + g is reducible for all g E GF(q)[x] with small degree. This is the analogue for polynomials of a result of Erdos and Rankin concerning gaps between consecutive primes.

The Identification of Non-linear Systems with Statistically Equivalent Polynomial Systems

Abstract: Volterra (or Wiener) series models of non-linear systems have been studied by many authors, e.g. Rugh[1], Tomlinson[2]. By means of the Multi-dimensional Frequency Response Function (MFRF) based on the Volterra (or Wiener) kernel the non-linear dynamical systems can be analysed in the frequency domain. However, there are some crucial problems concerning the estimation and application of MFRFs[3]. The first is the expense of estimating MFRFs by direct methods, which impels us to study the parameter estimation methods. Billings[4] has developed the estimation method of MFRF by means of the NARMAX model. The polynomial approach for estimating MFRF is discussed in detail in the paper. From the point of view of the estimation of MFRF, the NARMAX model looks more convenient because the output relates directly to the input. However, from the point of view of the discussion on dynamics behaviour, there are more advantages with the parameter identification of the polynomial non-linear model. This is because it is used together with the equation of motion of the dynamic system. Natke and Zamirowski[5] discussed the method of structure identification for the class of polynomials within mechanical systems, which laid a foundation for the parameter estimation method of MFRF. The second crucial problem is that the difficulties arise from using MFRF to describe the non-linear properties of the system because the multi-frequency is without a physical meaning. In the second part of this paper an attempt is made to decrease these difficulties. While the standardized formula of the parameter estimation is given with the response and the excitation, emphasis is given to the study of the statistical equivalent 3rd order polynomial system, the spectral structure of the response expressed with MFRF is analysed in detail, and the Extended Transfer Functions (ETF) are defined, which are only functions of one-dimension frequency. In addition, the effectiveness of the statistical equivalence is analysed. The advantages of the method used here are that the MFRF of the polynomial non-linear system has the theoretical analytical expression, and by studying a non-linear system particular properties with extended transform functions avoid the difficulty of graphing the MFRF.

Solving Systems of Polynomial Equations

Abstract: The Grobner basis algorithm can be seen to be a generalization of the classical Gaussian elimination algorithm from a set of linear multivariate polynomials to an arbitrary set of multivariate polynomials. The S-polynomial and reduction processes take the place of the pivoting step of the Gaussian algorithm. Taking this analogy much further, one can devise a constructive procedure to compute the set of solutions of a system of arbitrary multivariate polynomial equations: $$ \begin{array}{*{20}{c}} {{f_1}\left( {{x_1}, \ldots ,{x_n}} \right) = 0,} \\ {{f_2}\left( {{x_1}, \ldots ,{x_n}} \right) = 0,} \\ \vdots \\ {{f_r}\left( {{x_1}, \ldots ,{x_n}} \right) = 0,} \end{array} $$ i.e., compute the set of points where all the polynomials vanish: $$ \left\{\langle\xi_{1},\ldots,\xi_{n}\rangle:f_{i}(\xi_{1},\ldots,\xi_{n})=0,\quad{\rm for}\ {\rm all}\ 1\leq i\leq r\right\}. $$

A finite field arithmetic unit VLSI chip

Abstract: This paper presents a circuit operating on fields F2 [X] /〈f(X)〉 where f(X) is a binary irreducible polynomial of degree m ≥ 2, and F2=GF(2). This circuit is able to perform back to back multiplications and inversions for any such f(X) and any value of m within a specified range, m being possibly large. It is assumed that the elements of the field are expressed as polynomials in X of degree less than m (polynomial basis). The circuit consists mainly of a Serial Input-Serial Output multiplier which is similar to the one published by Yeh, Reed, Truong in 1984. An element of the field is inverted by raising it to the power 2 m — 2, and so the outputs of the multiplier are fed back into its inputs. Even though the circuit can operate on any size of field within the specified range, it is better suited for large fields; circuitry achieving better performance can be designed for small fields (Parallel Input-Parallel Output).

Graphs corresponding to reference polynomial or to circuit characteristic polynomial

Abstract: A vertex-weighted graphG* is studied which is obtained by deleting edgee rs in a circuit of a graphG and giving two vertices νr and es weightsh r = 1 andh s = -1, respectively It is shown that if subgraphG - νr is identical with subgraphG - νs, then the reference polynomial ofG* is identical with that ofG and the characteristic polynomial ofG* contains the contributions due to only a certain part of the circuits found in the original graphG This result gives a simple way to find a graph whose characteristic polynomial is equal to the reference polynomial in the topological resonance energy theory or to the circuit characteristic polynomial in the circuit resonance energy theory This approach can be applied not only to Hilckel graphs but also to Mobius graphs, provided that they satisfy a certain condition The significances of this new type of “reference” graph thus obtained are pointed out