Showing papers on "Square-free polynomial published in 2004"

QR factoring to compute the GCD of univariate approximate polynomials

Abstract: We present a stable and practical algorithm that uses QR factors of the Sylvester matrix to compute the greatest common divisor (GCD) of univariate approximate polynomials over /spl Ropf/[x] or /spl Copf/[x]. An approximate polynomial is a polynomial with coefficients that are not known with certainty. The algorithm of this paper improves over previously published algorithms by handling the case when common roots are near to or outside the unit circle, by splitting and reversal if necessary. The algorithm has been tested on thousands of examples, including pairs of polynomials of up to degree 1000, and is now distributed as the program QRGCD in the SNAP package of Maple 9.

Positive extensions, Fejér-Riesz factorization and autoregressive filters in two variables

Abstract: In this paper we treat the two-variable positive extension problem for trigonometric polynomials where the extension is required to be the reciprocal of the absolute value squared of a stable polynomial. This problem may also be interpreted as an autoregressive filter design problem for bivariate stochastic processes. We show that the existence of a solution is equivalent to solving a finite positive definite matrix completion problem where the completion is required to satisfy an additional low rank condition. As a corollary of the main result a necessary and sufficient condition for the existence of a spectral FejerRiesz factorization of a strictly positive two-variable trigonometric polynomial is given in terms of the Fourier coefficients of its reciprocal. Tools in the proofs include a specific two-variable Kronecker theorem based on certain elements from algebraic geometry, as well as a two-variable Christoffel-Darboux like formula. The key ingredient is a matrix valued polynomial that appears in a parametrized version of the Schur-Cohn test for stability. The results also have consequences in the theory of two-variable orthogonal polynomials where a spectral matching result is obtained, as well as in the study of inverse formulas for doubly-indexed Toeplitz matrices. Finally, numerical results are presented for both the autoregressive filter problem and the factorization problem.

The approximate GCD of inexact polynomials

Abstract: This paper presents an algorithm and its implementation for computing the approximate GCD (greatest common divisor) of multivariate polynomials whose coefficients may be inexact. The method and the companion software appear to be the first practical package with such capabilities. The most significant features of the algorithm are its robustness and accuracy as demonstrated in computational experiments. In addition, two variations of a squarefree factorization method for multivariate polynomials are proposed as an application of the GCD algorithm.

Approximate factorization of multivariate polynomials via differential equations

Abstract: The input to our algorithm is a multivariate polynomial, whose complex rational coefficients are considered imprecise with an unknown error that causes f to be irreducible over the complex numbers C. We seek to perturb the coefficients by a small quantitity such that the resulting polynomial factors over C. Ideally, one would like to minimize the perturbation in some selected distance measure, but no efficient algorithm for that is known. We give a numerical multivariate greatest common divisor algorithm and use it on a numerical variant of algorithms by W. M. Ruppert and S. Gao. Our numerical factorizer makes repeated use of singular value decompositions. We demonstrate on a significant body of experimental data that our algorithm is practical and can find factorizable polynomials within a distance that is about the same in relative magnitude as the input error, even when the relative error in the input is substantial (10-3).

An Abstract Interpretation Approach for Automatic Generation of Polynomial Invariants

Abstract: A method for generating polynomial invariants of imperative programs is presented using the abstract interpretation framework. It is shown that for programs with polynomial assignments, an invariant consisting of a conjunction of polynomial equalities can be automatically generated for each program point. The proposed approach takes into account tests in conditional statements as well as in loops, insofar as they can be abstracted to be polynomial equalities and disequalities. The semantics of each statement is given as a transformation on polynomial ideals. Merging of paths in a program is defined as the intersection of the polynomial ideals associated with each path. For a loop junction, a widening operator based on selecting polynomials up to a certain degree is proposed. The algorithm for finding invariants using this widening operator is shown to terminate in finitely many steps. The proposed approach has been implemented and successfully tried on many programs. A table providing details about the programs is given.

Complexity issues in bivariate polynomial factorization

Abstract: Many polynomial factorization algorithms rely on Hensel lifting and factor recombination. For bivariate polynomials we show that lifting the factors up to a precision linear in the total degree of the polynomial to be factored is sufficient to deduce the recombination by linear algebra, using trace recombination. Then, the total cost of the lifting and the recombination stage is subquadratic in the size of the dense representation of the input polynomial. Lifting is often the practical bottleneck of this method: we propose an algorithm based on a faster multi-moduli computation for univariate polynomials and show that it saves a constant factor compared to the classical multifactor lifting algorithm.

Inverse power and Durand-Kerner iterations for univariate polynomial root-finding

Abstract: Univariate polynomial root-finding is the oldest classical problem of mathematics and computational mathematics, and is still an important research topic, due to its impact on computational algebra and geometry. The Weierstrass (Durand-Kerner) approach and its variations as well as matrix methods based on the QR algorithm are among the most popular practical choices for simultaneous approximation of all roots of a polynomial. We propose an alternative application of the inverse power iteration to generalized companion matrices for polynomial root-finding, demonstrate its effectiveness, and relate its study to unifying the derivation of the Weierstrass (Durand-Kerner) algorithm (having quadratic convergence) and its extensions having convergence rates 4, 6, 8, …. Our experiments show substantial improvement versus the latter algorithm, even though the inverse power iteration is most effective for the more limited tasks of approximating a single root or a few selected roots.

Computing the RSA Secret Key Is Deterministic Polynomial Time Equivalent to Factoring

Abstract: We address one of the most fundamental problems concerning the RSA cryptoscheme: Does the knowledge of the RSA public key/ secret key pair (e,d) yield the factorization of N=pq in polynomial time? It is well-known that there is a probabilistic polynomial time algorithm that on input (N,e,d) outputs the factors p and q. We present the first deterministic polynomial time algorithm that factors N provided that e,d < φ(N) and that the factors p, q are of the same bit-size. Our approach is an application of Coppersmith’s technique for finding small roots of bivariate integer polynomials.

Limitations of Quantum Advice and One-Way Communication

Abstract: Although a quantum state requires exponentially many classical bits to describe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum messages have only limited advantages over classical ones. First, we show that $\mathsf{BQP/qpoly}\subseteq\mathsf{PP/poly}$, where $\mathsf{BQP/qpoly}$ is the class of problems solvable in quantum polynomial time, given a polynomial-size "quantum advice state" that depends only on the input length. This resolves a question of Buhrman, and means that we should not hope for an unrelativized separation between quantum and classical advice. Underlying our complexity result is a general new relation between deterministic and quantum one-way communication complexities, which applies to partial as well as total functions. Second, we construct an oracle relative to which $\mathsf{NP} ot \subset \mathsf{BQP/qpoly}$. To do so, we use the polynomial method to give the first correct proof of a direct product theorem for quantum search. This theorem has other applications; for example, it can be used to fix a result of Klauck about quantum time-space tradeoffs for sorting. Third, we introduce a new trace distance method for proving lower bounds on quantum one-way communication complexity. Using this method, we obtain optimal quantum lower bounds for two problems of Ambainis, for which no nontrivial lower bounds were previously known even for classical randomized protocols. A preliminary version of this paper appeared in the 2004 Conference on Computational Complexity (CCC).

A note on efficient computation of cube roots in characteristic 3.

Abstract: The cost of the folklore algorithm for computing cube roots in F3m in standard polynomial basis is less that one multiplication, but still O(m). Here we show that, if F3m is represented in trinomial basis as F3[x]/(x + ax + b) with a, b = ±1, the actual cost of computing cube roots in F3m is only O(m).

K-theory of truncated polynomial algebras

Abstract: 3.

Computing the RSA secret key is deterministic polynomial time equivalent to factoring

Abstract: We address one of the most fundamental problems concerning the RSA cryptoscheme: Does the knowledge of the RSA public key/ secret key pair (e, d) yield the factorization of N = pq in polynomial time? It is well-known that there is a probabilistic polynomial time algorithm that on input (N, e, d) outputs the factors p and q. We present the first deterministic polynomial time algorithm that factors N provided that e, d < Φ(N) and that the factors p, q are of the same bit-size. Our approach is an application of Coppersmith's technique for finding small roots of bivariate integer polynomials.

Algorithm 835: MultRoot---a Matlab package for computing polynomial roots and multiplicities

Abstract: MultRoot is a collection of Matlab modules for accurate computation of polynomial roots, especially roots with non-trivial multiplicities. As a blackbox-type software, MultRoot requires the polynomial coefficients as the only input, and outputs the computed roots, multiplicities, backward error, estimated forward error, and the structure-preserving condition number. The most significant features of MultRoot are the multiplicity identification capability and high accuracy on multiple roots without using multiprecision arithmetic, even if the polynomial coefficients are inexact. A comprehensive test suite of polynomials that are collected from the literature is included for numerical experiments and performance comparison.

Absolute polynomial factorization in two variables and the knapsack problem

Abstract: A recent algorithmic procedure for computing the absolute factorization of a polynomial P(X,Y), after a linear change of coordinates, is via a factorization modulo X3. This was proposed by A. Galligo and D. Rupprecht in [7],[16]. Then absolute factorization is reduced to finding the minimal zero sum relations between a set of approximated numbers b;i;, i =1 to n such that ∑n;i =1;b;i; =0, (see also [17]). Here this problem with an a priori exponential complexity, is efficiently solved for large degrees (n›100). We rely on LLL algorithm, used with a strategy of computation inspired by van Hoeij's treatment in [23]. For that purpose we prove a theorem on bounded integer relations between the numbers b;i;,, also called linear traces in [19].

Solving Nonlinear Polynomial Systems via Symbolic-Numeric Elimination Method

Abstract: Consider a general polynomial system S in x1, . . . , xn of degree q and its corresponding vector of monomials of degree less than or equal to q. The system can be written as M0 · [xq1, xq−1 1 x2, . . . , xn, x1, . . . , xn, 1] = [0, 0, . . . , 0, 0, . . . , 0, 0] (1) in terms of its coefficient matrix M0. Here and hereafter, [...] T means the transposition. Further, [ξ1, ξ2, . . . , ξn] is one of the solutions of the polynomial system, if and only if [ξ 1 , ξ q−1 1 ξ2, . . . , ξ 2 n, ξ1, . . . , ξn, 1] T (2) is a null vector of the coefficient matrix M0. Since the number of monomials is usually bigger than the number of polynomials, the dimension of the null space can be big. The aim of completion methods, such as ours and those based on Grobner bases and others [4, 5, 6, 7, 8, 10, 16, 18, 17, 12, 9, 20], is to include additional polynomials belonging to the ideal generated by S, to reduce the dimension to its minima. The bijection φ : xi ↔ ∂ ∂xi , 1 ≤ i ≤ n, (3) maps the system S to an equivalent system of linear homogeneous PDEs denoted by R. Jet space approaches are concerned with the study of the jet variety V (R) = {( u q , u q−1 , . . . , u 1 , u ) ∈ J : R ( u q , u q−1 , . . . , u 1 , u ) = 0 } , (4) where u j denotes the formal jet coordinates corresponding to derivatives of order exactly j. A single prolongation of a system R of order q consists of augmenting the system with all possible derivatives of its equations, so that the resulting augmented systems, denoted by DR, has order q + 1. Under the bijection φ, the equivalent operation for polynomial systems is to multiply by monomials, so that the resulting augmented system has degree q + 1. A single geometric projection is defined as E(R) := {( u q−1 , . . . , u 1 , u ) ∈ Jq−1 : ∃ u q , R ( u q , u q−1 , . . . , u 1 , u ) = 0 } . (5)

Does polynomial parallel volume imply convexity

Abstract: For a non-empty compact set A⊂ℝ d , d≥2, and r≥0, let A ⊕ r denote the set of points whose distance from A is r at the most. It is well-known that the volume, V d (A ⊕r ), of A ⊕r is a polynomial of degree d in the parameter r if A is convex. We pursue the reverse question and ask whether A is necessarily convex if V d (A ⊕r ) is a polynomial in r. An affirmative answer is given in dimension d=2, counterexamples are provided for d≥3. A positive resolution of the question in all dimensions is obtained if the assumption of a polynomial parallel volume is strengthened to the validity of a (polynomial) local Steiner formula.

Nonlinear polynomial systems: multiple roots and their multiplicities

Abstract: We present methods for the computation of roots of univariate and bivariate nonlinear polynomial systems as well as the identification of their multiplicity. We first present an algorithm, called the TDB algorithm, which computes the values and the multiplicities of roots of a univariate polynomial. The procedure is based on the concept of the degree of a certain Gauss map, which is deduced from the polynomial itself. In the bivariate case, we use a combination of resultants and our procedure for the univariate case, as the basis for developing an algorithm for locating the roots and computing their multiplicities. Our methods are robust and global in nature. Complexity analysis of the proposed methods is included together with comparison with standard subdivision methods. Examples illustrate our techniques.

Sparse-coefficient polynomial approximations for hardware implementations

Abstract: This paper presents a method for automatic generation of best polynomial approximations dedicated to hardware implementation. The generated polynomial approximations lead to high-speed and small hardware operators because of the use of sparse coefficients (i.e. we include fixed strings of zeros in the binary representation of the coefficients). Two different solutions have been investigated for the generation of the sparse-coefficient polynomial approximations. Our first results show up to 47% smaller coefficients compared to standard minimax approximations for comparable accuracy.

On the entire solutions of a polynomial vector field on C2

Abstract: We provide a classification of the polynomial vector fields on C 2 with a transcendental (non-algebraic and proper) entire solution. As a consequence of this it is proved that the completeness of a polynomial vector field X on a singular transcendental trajectory (that is, containing zeroes of X in its topological closure) implies that X is complete.

Constants of derivations in polynomial rings over unique factorization domains

Abstract: A well-known theorem, due to Nagata and Nowicki, states that the ring of constants of any K-derivation of K[x,y], where K is a commutative field of characteristic zero, is a polynomial ring in one variable over K. In this paper we give an elementary proof of this theorem and show that it remains true if we replace K by any unique factorization domain of characteristic zero.

Polynomial interpolation from multiples

Abstract: We are given an unknown polynomial f ∈ ℤ[x] by a black box which on input a ∈ ℤ returns a value rq · f(a) for some unknown nonzero rational numbers ra. If we have appropriate upper bounds on the numerator and denominator of ra and the degree of f, then the coefficients of f can be computed in probabilistic polynomial time.

Computational Approach to Polynomial Identities of Matrices – a Survey

Abstract: We present a survey on polynomial identities of matrices over a field of characteristic 0 from computational point of view. We describe several computational methods for calculation with polynomial identities of matrices and related objects. Among the other applications, these methods have been successfully used: The work of this author was partially supported by MURST of Italy. The work of this author was partially supported by Grant MM-1106/2001 of the Bulgarian Foundation for Scientific Research. (i) to show that all polynomial identities of degree 2n + 2 for the n× n matrix algebra for n = 3, 4, 5 are consequences of the standard identity s2n; (ii) to obtain upper bounds for the multiplicities of the irreducible S9-characters of the multilinear identities of degree 9 in 3×3 matrices; (iii) in the discovery of new central polynomials of low degree for matrices of order 3 and 4; (iv) to obtain upper bounds for the multiplicities of the irreducible S9-characters of ∗-polynomial identities of degree 9 in symmetric variables only for 6× 6 matrices with symplectic involution.

Inverse limits of polynomial rings

Abstract: In this note we deal with the question as to whether the inverse limit of onevariable polynomial rings is polynomial again. More precisely, working in the realm of the pro-affine algebra theory [1, 2], we look at the pro-affine algebra := lim← [ ] given over := lim← and ask if that algebra is isomorphic to [Y] for a suitable choice of variable Y. Our first answer is that is always locally polynomial over (see Th. 1), and our second answer is that ≃ [Y] if the inverse system of the units of ’s satisfies a ‘uniformized’ Mittag-Leffler condition (see Th. 2). After developing and proving these two theorems, we conclude this note with presentation of some examples in §3, originally made up by David Wright during the discussion sessions by him, N. Mohan Kumar and the present author in St. Louis, August 2002. Back in Japan in the fall of the same year the author was able to prove Theorem 2 through further study of these examples. The author wishes to record here his heartfelt thanks to the two friends just mentioned, as well as to M. Miyanishi who initially suggested the main question to us informally in July 2002 and to R.V. Gurjar together with whom the author made a first analysis of the question.