Showing papers on "Irreducible polynomial published in 1983"

Some Useful Bounds

Abstract: Some fundamental inequalities for the following values are listed: the determinant of a matrix, the absolute value of the roots of a polynomial, the coefficients of divisors of polynomials, and the minimal distance between the roots of a polynomial. These inequalities are useful for the analysis of algorithms in various areas of computer algebra.

Effective determination of the decomposition of the rational primes in a cubic field

Abstract: The decomposition of the rational primes in a cubic field K is determined in terms of the coefficients of a defining polynomial of K. As a consequence, the discrlminant D of K is straightforwardly computed and the cubic fields with index i(K) = 2 are easily characterized. Introduction. The study of the decomposition of the rational primes in a cubic field K is well known if K is cyclic [3, ?24] and the noncyclic case was developed by Hasse [5] and more recently by Martinet and Payan [6]. However, these results are not sufficient to determine such decompositions in terms of a defining polynomial of K. This is the problem we solve in this paper (Theorem 1). As a consequence, we obtain a straightforward computation of the discriminant of K (Theorem 2) and we reobtain in a completely elementary way a well-known theorem of Hasse [5, Theorem 6] on the possible values of the discriminant of a cubic field (Theorem 3). Finally, as another consequence of Theorem 1, we find a characterization of the cubic fields in which 2 is a common index divisor, which improves the given by Tornheim in [8, Theorem 2] (Theorem 4). Most of our results are easily extendible to the case of relative cubic extensions, but we will not make these generalizations explicit. Let K be a cubic field. We can suppose that K = Q(0), where 6 is a root of an irreducible polynomial of the type f(X) = X3-aX + b, a, b E Z. The discriminant of f(X) is A = 4a 327b2 and if we denote by D the discriminant of K we have A = i(9)2 D, where i(8) denotes the index of 9. For every prime p E Z and integer m E Z we denote by vp(m) the greatest exponent k such that pk j m. If for any prime p we have (1) vp(a) -2 and vp(b) 3, then 9/p is an algebraic integer whose equation is X3 (a/p2)X + (b/p 3). Therefore, we can assume that (1) is not satisfied for any prime p. Let s = vp(A) and AP = A/psp for every prime p. Received by the editors May 21, 1981. 1980 Mathematics Subject Classification. Primary 12A30. Kev words and phrases. Cubic field, ramification, discriminant, index of a number field. ?31983 American Mathematical Society 0002-9939/82/OO0-0768/$02.25

A Generalized Class of Polynomials that are Hard to Factor

Abstract: A class of univariate polynomials is defined which make the Berlekamp-Hensel factorization algorithm take an exponential amount of time. This class contains as subclasses the Swinnerton-Dyer polynomials discussed by Berlekamp and a subset of the cyclotomic polynomials. Aside from shedding light on the complexity of polynomial factorization this class is also useful in testing implementations of the Berlekamp–Hensel and related algorithms.

Some comments on the modular approach to Gröbner-bases

Abstract: The problem of finding a modular algorithm for constructing Grobner-bases is of interest to many computer algebraists. In particular, given a prime p and a set of (multivariate) polynomials with integer coefficients, it has been queried if the number of basis polynomials in a minimal normed Grobner-basis for the polynomial ideal generated mod p has to be less than or equal to the corresponding number for the polynomial ideal generated over the rationals. In this paper we answer this question and related questions concerning the modular approach to Grobner-bases, illustrating with several interesting examples, and we propose a criterion for determining "luckiness" of primes in the binomial case.

How to generate random integers with known factorization

Abstract: Recent work in public-key cryptography has led to the need to generate large random numbers with known factorization. This paper describes a probabilistic algorithm that produces a random k-bit integer in factored form. Each such number is equally likely to appear. The expected running time is, up to a constant factor, that required for k prime tests on k-bit integers. Thus, under reasonable assumptions about the speed of primality testing, it is a polynomial time process.

Polynomial-Time Factorization of Multivariate Polynomials over Finite Fields

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, ie 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 Also a deterministic version of the algorithm is discussed whose running time is polynomial in the degree of the input polynomial and the size of the field

Factoring multivariate polynomials over finite fields

Abstract: This paper describes an algorithm for the factorization of multivariate polynomials with coefficients in a finite field that is polynomial-time in the degrees of the polynomial to be factored. The algorithm makes use of a new basis reduction algorithm for lattices over IFq.

The rank of a commutator

Abstract: Let A ∊ Mn (D) with D a division ring. It is shown that unless A satisfies a central irreducible polynomial there exist Xk ∊Mn (D) so that rank (XkA − AXk ) = k for any k  2 min {rank(A − cI)∣c ∊ Z(D)}. The possible ranks of commutators are determined in the other case.

Parallel algorithms for arithmetics, irreducibility and factoring of GFq-polynomials

Abstract: A new algorithm for testing irreducibility of polynomials over finite fields without gcd computations makes it possible to devise efficient parallel algorithms for polynomial factorization. We also study the probability that a random polynomial over a finite field has no factors of small degree.

Factorisation of sparse polynomials

Abstract: Sparse polynomials xn±1 are often treated specially by the factorisation programs of computer algebra systems. We look at this, and ask how far this can be generalised. The answer is that more can be done for general binomials than is usually done, and recourse to a general purpose factoriser can be limited to "small" problems, but that general trinomials and denser polynomials seem to be a lost cause. We are concerned largely with the factorisation of univariate polynomials over the integers, being the simplest case.

Polynomial factorization by root approximation

Abstract: We show that a constructive version of the fundamental theorem of algebra [3], combined with the basis reduction algorithm from [1], yields a polynomial-time algorithm for factoring polynomials in one variable with rational coefficients.

On computing galois groups and its application to solvability by radicals

Abstract: This thesis presents a polynomial time algorithm for the basic question of Galois theory, checking the solvability by radicals of a monic irreducible polynomial over the integers It also presents polynomial time algorithms for factoring polynomials over algebraic number fields, for computing blocks of imprimitivity of roots of a polynomial under the transitive action of number fields (In all of these algorithms it is assumed that the algebraic number field is given by a primitive element which generates it over the rationals, and that the polynomial in question is monic, with coefficients in the integers) We also show how to express a root in radicals in terms of a straight line program in polynomial time The techniques used include methods from computational complexity and approaches from the theory of finite permutation groups The results presented here rely on the recent work of Lenstra, Lenstra, and Lovasz, in which a polynomial time algorithm for factoring polynomials over the integers is presented Many questions remain Our divide-and-conquer approach answers the question of solvability without revealing the nature of the group in question; we do not even learn its order We suggest this as one of the many open problems that remain to be tackled

The Complexity of Solving Polynomial Equations by Prime Root Extractions

Abstract: We show that for each algebraic number field Q of finite degree and for each natural number $n = p_1 \cdots p_k ,p_i $ prime, there exists an irreducible polynomial $f_n $ of degree n in $Q[ x ]$ such that $f_n $ is solvable by radicals and $1 + p_1 + p_1 p_2 + \cdots + p_1 p_2 \cdots p_{k - 1} $ extractions of $p_i $th roots are required for obtaining all roots of $f_n $. This generalizes the linear lower bound on the number of circles one has to draw in order to solve a geometric construction problem by ruler and compass and exponentially improves the obvious lower bound which is the number of prime factors of $\varphi ( n )$, the value of Euler’s function on n.

The SO(7) polynomial basis for symmetric representations

Abstract: A polynomial basis is derived for the symmetric irreducible representations of the group SO (7). The reduction of SO(7) into [SU(2)]3 is considered. The SO(7) generators not belonging to [SU(2)]3 are grouped into a bispinor vector, of which matrix elements are calculated. An explicit expression for the state vector is given.