Showing papers on "Square-free polynomial published in 1985"
TL;DR: An algorithm is presented which reduces the problem of finding the irreducible factors of f in polynomial-time in the total degree of f and the coefficient lengths of f to factoring a univariate integral polynomials, which implies the following theorem.
Abstract: Consider a polynomial f with an arbitrary but fixed number of variables and with integral coefficients. We present an algorithm which reduces the problem of finding the irreducible factors of f in polynomial-time in the total degree of f and the coefficient lengths of f to factoring a univariate integral polynomial. Together with A. Lenstra’s,.H. Lenstra’s and L. Lovasz’ polynomial-time factorization algorithm for univariate integral polynomials [Math. Ann., 261 (1982), pp. 515–534] this algorithm implies the following theorem. Factoring an integral polynomial with a fixed number of variables into irreducibles, except for the constant factors, can be accomplished in deterministic polynomial-time in the total degree and the size of its coefficients. Our algorithm can be generalized to factoring multivariate polynomials with coefficients in algebraic number fields and finite fields in polynomial-time. We also present a different algorithm, based on an effective version of a Hilbert Irreducibility Theorem, w...
131 citations
TL;DR: In this paper, the 1963 Davis algorithm for the spectral factorization of a nonnegative polynomial matrix is revised by introducing the notion of diagonal reducedness of the matrix, which is an easy sufficient test for the absence of zeros at infinity.
Abstract: We revise the 1963 Davis algorithm [2] for the spectral factorization of a para-Hermitian nonnegative polynomial matrix \Phi , by symmetric factor extraction: this algorithm is careless about zeros at infinity. By introducing the notion of diagonal reducedness of \Phi , we obtain an easy sufficient test for the absence of zeros at infinity. We show then how to get \Phi , diagonally reduced by diagonal excess reduction steps (similar to the Oono and Yasuura steps), removing all zeros at infinity, and then how to remove synunetrically finite zeros while keeping el, diagonally reduced (hence, free of zeros at infinity). This results in a revised symmetric extraction spectral factorization algorithm with monotone degree control. An example shows the didactical conceptual simplicity of the method. Appropriate symmetric extraction is discovered by revising and discovering important particular one-sided factor extraction properties of polynomial matrices.
114 citations
01 Apr 1985
TL;DR: It is shown how the content of the input polynomial in the main variable as a by-product can be taken advantage of when computing the GCD of multivariate polynomials by sparse Hensel lifting.
Abstract: A new algorithm is introduced which computes the multivariate leading coefficients of polynomial factors from their univariate images This algorithm is incorporated into a sparse Hensel lifting scheme and only requires the factorization of a single univariate image The algorithm also provides the content of the input polynomial in the main variable as a by-product We show how we can take advantage of this property when computing the GCD of multivariate polynomials by sparse Hensel lifting
69 citations
TL;DR: A probabilistic algorithm that finds the irreducible factors of a bivariate polynomial with coefficients from a finite field in timePolynomial in the input size, i.e., in the degree of the polynomials and log (cardinality of field).
Abstract: We present a probabilistic algorithm that finds the irreducible factors of a bivariate polynomial with coefficients from a finite field in time polynomial in the input size, i.e., in the degree of the polynomial and log (cardinality of field). The algorithm generalizes to multivariate polynomials and has polynomial running time for densely encoded inputs. A deterministic version of the algorithm is also discussed, whose running time is polynomial in the degree of the input polynomial and the size of the field.
55 citations
51 citations
TL;DR: In this article, the problem of assigning characteristic polynomials by dynamic output feedback in 2D systems is solved via 2D polynomial methods and a simple necessary and sufficient condition is given.
Abstract: The problem of characteristic polynomial assignment by dynamic output feedback in 2-D systems is solved via 2-D polynomial methods. A simple necessary and sufficient condition is given. The class of all assignable characteristic polynomials is described in a parametric form. Stable and causal solutions are treated.
28 citations
TL;DR: In this article, the relation between polynomial addition sets and Cayley x n -digraphs and strongly regular Cayley graphs is studied, and a complete list of polynomials with certain restrictions on parameters is given.
14 citations
TL;DR: In this article, it was shown that if all the zeros of P(z) lie in |z| < 1 and W1, W2, W3, W4, W5, Wn−1 are the zero points of P′(z), then each of the disks |(z/2) −wj| < ½ and |z-Wj| = 1, j = 1/2, 2, …, n−1 contains at least one zero of p(z).
Abstract: Let P(z) be a polynomial of degree n and P′(z) be its derivative. Given a zero of P′(z), we shall determine regions which contains at least one zero of P(z). In particular, it will be shown that if all the zeros of P(z) lie in |z| < 1 and W1, W2, …, Wn−1 are the zeros of P′(z), then each of the disks |(z/2)–wj| < ½ and |z–Wj| < 1, j = 1, 2, …, n−1 contains at least one zero of P(z). We shall also determine regions which contain at least one zero of the polynomials mP(z) + zP′(z) and P′(z) under some appropriate assumptions. Finally some other results of similar nature will be obtained.
13 citations
TL;DR: In this paper, the authors demonstrated the possibility of using a bivariate factorization algorithm in phase retrieval when a finite degree polynomial model is adopted to describe the (at least twice over-sampled) intensity function.
Abstract: In this paper is demonstrated the possibility of using a bivariate factorization algorithm in phase retrieval when a finite degree polynomial model is adopted to describe the (at least twice over-sampled) intensity function.
11 citations
TL;DR: In this paper, the authors unify the proofs of these two facts in results that they hope will be helpful in identifying the pole assignability property (or its absence) in other rings.
11 citations
TL;DR: Cooperativity in the protein-ligand binding process is discussed in terms of the zeros of the binding polynomial and the corresponding possible factorizations of the binders into polynomials having non-negative coefficients.
TL;DR: In this paper, the concept of a multiple root of matrix polynomial L (λ) is introduced, and associated spectral properties of L(λ) are investigated, and applications are made to factorizations of L α (λ), for any positive integer α.
TL;DR: The Bezoutian B of two polynomial matrices can be described as a solution of a linear matrix equation as discussed by the authors, which yields a new proof of the Barnett factorization of B.
TL;DR: It is shown the near-universal existence of a second invariant that is polynomial in the momenta for integrable Hamiltonian systems in two dimensions.
Abstract: We show the near-universal existence of a second invariant that is polynomial in the momenta for integrable Hamiltonian systems in two dimensions. Specifically, Hietarinta's three ''counterexamples'' are converted to polynomial form.
01 Jan 1985
TL;DR: In this paper, the authors define a circular domain as a domain in ℂ whose boundary is a circle or a straight line, and by D(a;η) its closure.
Abstract: By a “circular domain” we shall mean a domain in ℂ whose boundary is a circle or a straight line. By D(a;η) we shall denote the open disk {z ∈ ℂ : ∣z−a∣< η} and by D(a;η) its closure.
Proceedings Article•
01 Jun 1985TL;DR: The design for a "Polynomial Transformer" which executes the transformation of any relations among finite fields to a unique polynomial of one variable using Galois Fields is explained.
Abstract: Any relations among finite fields can be transformed to a unique polynomial of one variable using Galois Fields. In this paper, we explain the design for a "Polynomial Transformer" which executes the transformation. Polynomial Transformer consists of very simple and iterative logic, and it is very suitable for parallel and pipelined VLSI algorithm. Moreover, three dimensional construction of a Polynomial Transformer is possible. Thus, it serves as an example of a typical three dimensional VLSI. Its application can be found in Polynomial Transformation, disturbance of data and so on.
TL;DR: In this paper, an algorithm is proposed which combines global as well as local convergences, i.e., it ensures a rapid hit into a small neighbourhood of a root for 2-3 iterations from any random initial approximation and cubic convergence within the neighbourhood.
Abstract: An algorithm is suggested which performs fast calculations of all the roots of a polynomial with maximal computer accuracy using, as the only primary information, the coefficients and the degree of the polynomial. The algorithm combines global as well as local convergences, i.e. it ensures a rapid hit into a small neighbourhood of a root for 2–3 iterations from any random initial approximation and cubic convergence within the neighbourhood. A modification of the method is given which allows the roots of entire functions to be found. A numerical comparison of the method with commonly used methods (Newton–Raphson, Bairstow–Newton, steepest descent) shows its advantages in speed, accuracy and stability both for real and complex polynomials.
TL;DR: In this article, a method is presented for decomposing a rational function into the sum of the K partial fraction terms which proceed from the repeated polynomial plus a proper rational function which completes the equality.
Abstract: Attention is directed to those proper rational functions whose denominators may be expressed as the product of an Nth degree polynomial raised to the Kth power and another polynomial of degree M. A method is presented for decomposing such a rational function into the sum of the K partial fraction terms which proceed from the repeated polynomial plus a proper rational function which completes the equality. Use is made of an extended version of Homer's scheme. Two numerical examples and an operations count are presented. The method is free of complex arithmetic provided that all of the coefficients of the entering polynomials are real. Introduction. Consider the proper rational function (1) F(s) = -KB (s) QK(S )A(s)' where Q(s) is a polynomial of degree N (N > 1), A(s) is a polynomial of degree M which shares none of its zeros with Q(s); B(s) is a polynomial of degree m (m 1 by writing QK(S) as Q(s), which has the effect of redacing the problem to the K = 1 case. For this problem one may find C1(s) which, in Lurn, by repeated use of long division, may be written as CI(s) + C2(s)Q(s) + _K* S -F C(s)QK(s). Subsequent division by QK(S) gives the summation portion of (2). Received March 8, 1983; revised November 16, 1983 and February 10, 1984. 1980 Mathematics Subject Classification. Primary 65F99. ?1985 American Mathematical Society 0025-5718/85 $1.00 + $.25 per page 167 This content downloaded from 157.55.39.58 on Tue, 11 Oct 2016 04:48:47 UTC All use subject to http://about.jstor.org/terms
TL;DR: In this article, a determinant form formula for the general solution of coupled linear equations with coefficients in K[XI,....... xn], where K is a field of numbers, the number of unknowns is greater than the number number of equations, and the solutions are in K(x\,..., Xn-i)[xn]
Abstract: This paper derives a determinant form formula for the general solution of coupled linear equations with coefficients in K[XI, ....... xn], where K is a field of numbers, the number of unknowns is greater than the number of equations, and the solutions are in K(x\, ..., Xn-i)[xn]The formula represents the general solution by the minimum number of generators, and it is a generalization of Cramer's formula for the solutions in K ( X I , ..., xn)Compared with another formula which is obtained by a method typical in algebra, the generators in our formula are represented by determinants of quite small orders. §
TL;DR: This paper gives a new method of factorization of a polynomial P over ℤ, grounded on the fact, that any squarefree polynometric has a simple p-adic root, using Newton's method.
Abstract: This paper gives a new method of factorization of a polynomial P over ℤ. The method is grounded on the fact, that any squarefree polynomial has a simple p-adic root. The algorithm starts from a simple root of P over ℤ/pℤ and from this root the algorithm computes the corresponding root of P over ℤ/pk ℤ, using Newton's method. So we obtain a linear factor of P.
01 Apr 1985
TL;DR: In this paper, the multipliers of polynomial addition sets are studied and a complete characterization of the case where G is cyclic and f ( x ) is irreducible is given.