Showing papers on "Square-free polynomial published in 1997"

Small Solutions to Polynomial Equations, and Low Exponent RSA Vulnerabilities

Abstract: We show how to find sufficiently small integer solutions to a polynomial in a single variable modulo N, and to a polynomial in two variables over the integers. The methods sometimes extend to more variables. As applications: RSA encryption with exponent 3 is vulnerable if the opponent knows two-thirds of the message, or if two messages agree over eight-ninths of their length; and we can find the factors of N=PQ if we are given the high order $\frac{1}{4} \log_2 N$ bits of P.

Characteristic Sets for Polynomial Grammatical Inference

Abstract: When concerned about efficient grammatical inference two issues are relevant: the first one is to determine the quality of the result, and the second is to try to use polynomial time and space. A typical idea to deal with the first point is to say that an algorithm performs well if it infers {\it in\ the\ limit} the correct language. The second point has led to debate about how to define polynomial time: the main definitions of polynomial inference have been proposed by Pitt and Angluin. We return in this paper to a definition proposed by Gold that requires a characteristic set of strings to exist for each grammar, and this set to be polynomial in the size of the grammar or automaton that is to be learned, where the size of the sample is the sum of the lengths of all strings it includes. The learning algorithm must also infer correctly as soon as the characteristic set is included in the data. We first show that this definition corresponds to a notion of teachability as defined by Goldman and Mathias. By adapting their teacher/learner model to grammatical inference we prove that languages given by context-free grammars, simple deterministic grammars, linear grammars and nondeterministic finite automata are not identifiable in the limit from polynomial time and data.

A reordered Schur factorization method for zero-dimensional polynomial systems with multiple roots

Abstract: The technique of solving systems of multivariate polynomial equations via rigenproblems has become a topic of active research (with applications in computer-aided design and {untrul theory, for example) at least since the papers [2, 6, 9]. one may approach the problem via various resultant formulations ,x by Grijbner bases. As more understanding is gained, it is becoming clearer that eigenvalue problems are the “weakly nonlinear nucleus to which the original, strongly nonlinear task may be reduced’ [13], Earl,v works mmcemt,rat,ed on the case of simple roots. An example of such was, the paper [5], which used a numerical adaptation of il resultant technique due to Lazard to attack the problem directly, without, reference to Grobner bases.

Cyclic polytopes and the K -theory of truncated polynomial algebras

Abstract: This paper calculates the relative algebraic K -theory K∗(k [x ]/(x n ), (x )) of a truncated polynomial algebra over a perfect field k of positive characteristic p. Since the ideal generated by x is nilpotent, we can apply McCarthy’s theorem: the relative algebraic K -theory is isomorphic to the relative topological cyclic homology, [Mc], and it is the latter groups we actually evaluate. The result is best expressed in terms of big Witt vectors. Let Wm (k ) denote the big Witt vectors in k of length m , i.e. the multiplicative group Wm (k ) = (1 + xk [[x ]])×/(1 + x k [[x ]])×,

Covering points in the plane by k-tours: towards a polynomial time approximation scheme for general k

Abstract: Let P be a set of n points in the plane. A k-tour through P is a tour in the plane that starts and ends at the fixed origin and visits at most k points of P. Our goal is to cover all the points of P by k-tours so as to minimize the total length of the tours. We give a polynomial time approximation scheme (PTAS) for this problem whose running time is (k/~)Oikf’sl + O(n log n). Thus, our scheme remains a PTAS when k has a nearly logarithmic dependence on n. This improves over the previous best known scheme which is a PTAS only when k = O(log log n).

Fast polynomial factorization over high algebraic extensions of finite fields

Abstract: New algorithms are presented for factoring polynomials of degree n over the finite field of q elements, where q is a power of a fixed prime number. When log q = n, where a > 0 is constant, these algorithms are asymptotically faster than previous known algorithms, the fastest of which required time Ω(n(log q)), or Ω(n) in this case, which corresponds to the cost of computing x modulo an n degree polynomial. The new algorithms factor an arbitrary polynomial in time O(n+n). All measures are in fixed precision operations, that is in bit complexity. Moreover, in the special case where all the irreducible factors have the same degree, the new algorithms run in time O(n). In particular, one may test a polynomial for irreducibility in O(n) bit operations. These results generalize to the case where q = p, where p is a small prime number relative to q.

Polynomial Time Approximation Schemes for Some Dense Instances of NP-Hard Optimization Problems

Abstract: We survey recent results on the existence of polynomial time approximation schemes for some dense instances of NP-hard combinatorial optimization problems. We indicate some inherent limits for the existence of such schemes for some other dense instances of optimization problems. We also go beyond the dense optimization problems and show how other approximation problems can be solved by using dense techniques.

Generalized inverses of two-variable polynomial matrices and applications

Abstract: The main contribution of this paper is to present (a) an algorithm for the computation of the generalized inverse of a not necessarily square two-variable polynomial matrix and (b) some applications of the proposed algorithm to the solution of Diophantine equations

Stabilization of polynomial systems solving with Groebner bases

Abstract: Hans J. Stet ter Institute for Applied and Numerical Mathematics Technical University of Vienna stetter@uranus. tuwien. ac. at http: //info. tuwien. ac. at/tuwien/whitedsg .htm Groebner bases of O-dimensional polynomial ideals depend discontinuously on the coefficients of a generating set; for a specified term order, the location of these discontinuities relates to the location of the zero set of the ideaf. For a polynomial system with zeros near such a singularityy manifold, the Groebner basis is ill-suited for the numerical computation of the zeros. For this case, we propose an alternate representation (extended Groebner bases) which permits a stable computation of the zeros.

A note on GK dimension of skew polynomial extensions

Abstract: Let A be a finitely generated commutative domain over an algebraically closed field k, σ an algebra endomorphism of A, and δ a σ-derivation of A. Then GKdim(A[x, σ, δ]) = GKdim(A) + 1 if and only if σ is locally algebraic in the sense that every finite dimensional subspace of A is contained in a finite dimensional σ-stable subspace. Similarly, if F is a finitely generated field over k, σ a k-endomorphism of F , and δ a σ-derivation of F , then GKdim(F [x, σ, δ]) = GKdim(F ) + 1 if and only if σ is an automorphism of finite order.

Robust assignment of polynomial matrix polytopes

Abstract: Assignment of the coefficients of factors of the characteristic polynomial of an uncertain multi-input multi-output system to polytopic regions is considered. Owing to a special upper triangular form of the desired closed-loop polynomial matrix, this formulation leads to an affine design problem. This design is closer to the ultimate pole placement than the assignment of the characteristic polynomial direct. As a result, one obtains a set of affine inequalities with respect to the controller coefficients. The solution (if it exists) of these inequalities then defines all admissible controllers.

Linear approach to the least-squares multidimensional polynomial fitting

Abstract: It is well known that the least-squares one-dimensional (1-D) polynomial fitting problem is linear. However, multidimensional (M-D) polynomial fitting is often treated as a nonlinear problem. This paper shows that the least-squares M-D polynomial fitting problem is also a linear problem, and proposes a linear method for solving it. Two fitting examples in the 2-D case are given to illustrate the effectiveness of the proposed method.

ON THE LARGEST DEGREE OF AN IRREDUCIBLE FACTOR OF A POLYNOMIAL IN Fq(X)

Abstract: where Sf := deg f , is applied. We mention here the investigations [1, 5, 7-13, 17-20, 25]. On the other hand, there exists a parallel theory investigating the value distribution of the maps Sn ~ R, where Sn denotes the symmetric group of order n, when a permutation o" E S, is taken with the equal probability 1/n ! (see, for instance, [3, 6, I0, 12, 14, 21, 23, 26]). Observe that despite the fact that the same analytic or probabilistic methods can be applied, the problems arising in these two theories have been considered separately. To demonstrate a new point of view, we quote a corollary of Theorem 2 of the second author's paper [16]. Let p denote an irreducible polynomial, p e F~[X].

A New Approach to the Modal Regulator Synthesis for Interval Plant with Scalar Input

Abstract: We consider the problem of a closed-loop interval dynamical system stabilisation by a modal regulator which locates all coefficients of the open-loop system characteristic polynomial within assigned intervals. We present an original method for the modal regulator synthesis. This method differs from the analytic method [6] by the smaller number of operations with polynomial matrices. Also the method does not use the system canonical transformation as it is realized in [3], [11].

Fast parallel computation of the polynomial shift

Abstract: Given an n-degree polynomial f(x) over an arbitrary ring, the shift of f(x) by c is the operation which computes the coefficients of the polynomial f(x+c). In this paper, we consider the case when the shift by the given constant c has to be performed several times (repeatedly). We propose a parallel algorithm that is suited to an SIMD architecture to perform the shift in O(1) time if we have O(n/sup 2/) processor elements available. The proposed algorithm is easy to generalize to multivariate polynomial shifts. The possibility of applying this algorithm to polynomials with coefficients from non-commutative rings is discussed, as well as the bit-wise complexity of the algorithm.

A library of matlab functions for polynomial matrices

Abstract: This paper initiates the work pertaining to the preparation of a library of MATLAB functions. This library involves programs for personal computers, which realize methods and algorithms for solving spectral problems for polynomial one-parameter matrices, as well as some spectral problems for two-parameter polynomial matrices. Bibliography:4 titles.

An Inequality About the Largest Roots of a Polynomial

Abstract: We present an upper bound for the product of the k largest roots of a monic polynomial of degree n with complex coefficients. For that, we use the theory of compound matrices and the localization of the eigenvalues of matrices.

The average number of modular factors in Trager's polynomial factorization algorithm

Abstract: Trager’s algorithm for factoringa univariate polynomialover an algebraic number field computes thenorm of the polynomial and then factors the norm over the integers. It has been observed by the author as well as others that the norm tends to factor modulo a prime into more irreducible factors than one would expect from a typical random polynomial, but no explanation haa previously been given. In this paper, an exact formula is derived for the average number of irreducible factors of the norm modulo a prime. For large primes, the asymptotic average is larger than the corresponding average for random polynomials with integer coefficients.

Using Maple to solve multivariable central analysis and design problems for systems represented by polynomial matrices

Abstract: A collection of algorithms developed in a computer algebra package (MapleV) using polynomial matrix theory is described. The developed algorithms provide a medium in which polynomial matrix operations are carried out. Most importantly, these polynomial matrix procedures enable the design and analysis of multivariable control systems using the algebraic or polynomial equation approach. This algebraic design would have been extremely difficult to carry out in a strict numeric computing environment. The use of MapleV has provided symbolic results quickly and efficiently, with a tremendous gain in time and with minimal effort.

The Kalman-Yakubovich-Popov lemma in a behavioural framework and polynomial spectral factorization

Abstract: The classical Kalman-Yakubovich-Popov lemma provides a link between dissipativity of a system in state-space form and the solution to a linear matrix inequality. In this paper we derive the KYP lemma for linear systems described by higher-order differential equations. The result is an LMI in terms of the original coefficients in which the dissipativity problem is posed. Subsequently we study the connection between dissipativity and spectral factorization of polynomial matrices. This enables us to derive a new algorithm for polynomial spectral factorization in terms of an LMI in the coefficients of the polynomial matrix.