Showing papers on "Square-free polynomial published in 1982"

Factoring Polynomials with Rational Coefficients

Abstract: In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q(X). It is well known that this is equivalent to factoring primitive polynomials feZ(X) into irreducible factors in Z(X). Here we call f~ Z(X) primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. (8). Its running time, measured in bit operations, is O(nl2+n9(log(fD3).

Polynomial matrix primitive factorization over arbitrary coefficient field and related results

Abstract: Morf, Levy, and Kung and Youla and Gnavi presented a primitive factorization algorithm which extracts in some sense the content of a (full rank) matrix A with entries in the ring K[z,\omega] of bivariate polynomials over some field K . However, the algorithms presented in both cases specify and require the coefficient field K to be algebraically closed-typically the field of complex numbers. It is desirable, from theoretical and computational standpoints, to have no such restriction on K ; so, for example, one could do the factorization over the real field or even the field of rational numbers, provided the coefficients start out in these fields. Here an algorithm which produces a primitive factorization over an arbitrary field K is presented and the use of this algorithm is illustrated by a nontrivial example. Several related results leading to a general factorization theorem are stated and proved. Scopes for applying the results in various problems of scientific and engineering interest are mentioned.

A polynomial-time reduction from bivariate to univariate integral polynomial factorization

Abstract: An algorithm is presented which reduces the problem of finding the irreducible factors of a bivariate polynomial with integer coefficients in polynomial time in the total degree and the coefficient lengths to factoring a univariate integer polynomial. Together with A. Lenstra's, H. Lenstra's and L. Lovasz' polynomial-time factorization algorithm for univariate integer polynomials and the author's multivariate to bivariate reduction the new algorithm implies the following theorem. Factoring a polynomial with a fixed number of variables into irreducibles, except for the constant factors, can be accomplished in time polynomial in the total degree and the size of its coefficients. The new algorithm can be generalized to reducing multivariate factorization directly to univariate factorization and to factoring multivariate polynomials with coefficients in algebraic number fields and finite fields in polynomial time.

A Branching Process Arising in Dynamic Hashing, Trie Searching and Polynomial Factorization

Abstract: We obtain average value and distribution estimates for the height of a class of trees that occurs in various contexts in computer algorithms : in trie searching, as index in several dynamic schemes and as an underlying partition structure in polynomial factorization algorithms. In particular, results given here completely solve the problem of analyzing Extendible Hashing for which practical conclusions are given. The treatment relies on the saddle point method of complex analysis which is used here for extracting coefficients of a probability generating function, and on a particular technique that reveals periodic fluctuations in the behaviour of algorithms which are precisely quantified.

Polynomial algorithms for estimating network reliability

Abstract: We consider the problem of calculating the best possible bounds on the reliability of a system given limited information about the joint density function of its components. We show that a polynomial algorithm for this problem exists iff such an algorithm exists for a certain related problem of minimizing a linear objective function over a clutter. We give numerous examples of network as well as other problems for which the algorithm runs in polynomial time. We also use our construction to prove NP-hardness for others.

Stability Theorems for Overrings of Polynomial Rings

Abstract: Let R be a polynomial ring over a field. Projective modules over rings of the type R IX, Y] / ( r -X Y) where r is a non-zero element of R have been considered by Murthy, Swan, Weibel (see [Mu] and [W]). These rings lie between R[X] and R[ X ,X 1] and even when they are regular, projective modules over them need not be free ([Mu], Example 6.2). This leads us to study stability properties of projective modules over rings A which lie between R [X] and R[X,X-1] when R is any commutative noetherian ring. We prove stability theorems for projective modules over such rings A in w These results (Theorems 4.2 and 4.3) have been proved for A=R[X] by Plumstead [P] and for A = R [ X , X -1] by Mandal [Ma]. In w we prove similar results for the allied class of rings D of the type R[X, Y]/(XY). Stability theorems for GL,(A) and GL,(D) are proved in w When A=R[X] or R[X,X-1], our Theorems 6.2(i) and 6.4(i) had already been proved by Suslin [-S]. Finally in w 7 we study Pic(R[X,Y]/(r-XY)) when R is a PID. Theorem 7.1 extends a result of Murthy ([Mu], Corollary 5.3). It would be interesting to know whether Theorems 4.1 and 4.2 can be extended to the polynomial ring A[T 1 ..... Tm] (where A lies between R[X] and R[X,X-1]) . More precisely, for a projective A[T 1 .... ,T,,]-module P of rank > dim A we would like to know if the following statements are true: (i) P has a unimodular element. (ii) P has the cancellation property. We have been able to show that when A = R [ X ] , the statement (i) is true ([B-R], Theorem 3.1).

Stability of a matrix polynomial in discrete systems

Abstract: Two sufficient conditions that the determinant of a nonsingular real ( m \times m ) matrix polynomial of n th order has all its roots inside the unit circle have been obtained. These conditions are represented in terms of rational functions of the coefficient matrices. Therefore, these conditions do not require the computation of the determinant polynomial. The first condition is given in terms of the positive definiteness of an ( mn \times mn ) symmetric matrix, while the second condition is expressed by the positive definiteness of an ( m \times m ) Hermitian matrix which is a function of z, |z| \leq 1 . The first condition implies the second, and hence is more restrictive than the second.

Feedback characteristic polynomial controller design of 3-D systems in state-space

Abstract: For a general state-space model of three-dimensional (3-D) systems the characteristic polynomial (eigenvalue) control problem via state and output feedback is considered. A frequency domain approach is employed which in the scalar input case leads to a set of necessary and sufficient conditions. The multi-input problem is treated by assuming that the state or output feedback gain matrix is expressed as the dyadic product ⊙F = ⊙ ⊙f T of a column vector ⊙β and a row vector ⊙f T . This assumption leads to an equivalent scalar input problem β which is directly solved by using the scalar input results. Concerning the dynamic feedback compensator design problem, the important particular case of proportional plus integral plus derivative (PID) control is considered and treated by essentially the same algorithm, which leads to a linear algebraic system in the unknown parameters, along with some constraint equations upon the closed-loop characteristic polynomial sought.

On Polynomial Factorization

Abstract: These algorithms are probabilistic in the following sense. The time of computation depends on random choices, but the validity of the result does not depend on them. So, worst case complexity, being infinite, is meaningless and we compute average complexity. It appears that our and Cantor-Zassenhaus algorithms have the same asymptotic complexity and they are the best algorithms actually known ; with elementary multiplication and GCD computation, CantorZassenhaus algorithm is always better than ours ; with fast multiplication and GCD, it seems that ours is better, but this fact is not yet proveen.

Integral polynomial generators for the homology of

Abstract: Explicit formulas are given for polynomial generators of H*BSU as specific polynomials in the canonical polynomial generators of H*BU. The method is also applied to H*(BSU; R) for any coefficient ring R and to H*(BSO; Z2).

Exact computation of the greatest common divisor of two polynomial matrices

Abstract: Computation of G(s), the greatest common divisor GCD of two polynomial matrices D(s) and C(s) is of vital importance in the frequency domain approach to multivariable control systems. It is useful in the problem of nonsingular factorization of a polynomial matrix, minimal state-space realization of a rational function transfer matrix, relative primeness test of two polynomial matrices and so on. The problem has been tackled by many authors and through different techniques. An indirect method is to find an irreducible representation by any known algorithm in this field and then return to find out the GCD (see, e.g., EMRE [3J). There are other techniques to find the GCD as a polynomial combination, i.e., G(s) = P(s) . C(s)+Q(s)' D(s). (see, e.g., McDuFFEE [8J), or to transform the composite matrix [D'(s) C(s)]' to its upper-right triangular form [G'(s) O'J (see e.g., WOLOVICH [9J). The most significant method seems to be the extension of the well-known Sylvester's matrix of two scalar polynomials to the matrix case to form the so-called generalized Sylvester's matrix (see, e.g., ANDERSON [lJ and BITMEAD [2J). Neither of the methods mentioned above guarantee numerical stability. So it was suggested to use p-adic arithmetic to compute the GCD of two polynomial matrices by the generalized Sylvester's matrix method. Appendix A contains a brief discussion of p-adic arithmetic while the routines used to handle p-adic objects are listed in Appendix B. Definitions necessary to the GCD problem are given in chapter 2. Chapter 3 describes the algorithm and the main theory. An example is solved in chapter 4.

Solution of the equation (sI-A)R(s) + BQ(s) = I

Abstract: The polynomial equation (sI-A)R(s) + BQ(s) = In with (A, B) in the multicompanion controllable form is considered. The solution {R(s), Q(s)} is then explicitly determined, and, as an application, the transformation of certain well known polynomial matrices is given.