Showing papers on "Irreducible polynomial published in 1981"

Asymptotically fast factorization of integers

Abstract: The paper describes a "probabilistic algorithm" for finding a factor of any large composite integer n (the required input is the integer n together with an auxiliary sequence of random numbers). It is proved that the expected number of operations which will be required is O(exp{ 83Qn n In In n)l/2)) for some constant f > 0. Asymptotically, this algorithm is much faster than any previously analyzed algorithm for factoring integers; earlier algorithms have all required O(na) operations where a > 1/5.

Groups whose irreducible representations have finite degree

Abstract: Throughout this paper F denotes a (commutative) field. Let XF denote the class of all groups G such that every irreducible FC-module has finite dimension over F. In (1) P. Hall showed that if F is not locally finite and if G is polycyclic, then G∈XF if and only if G is abelian-by-finite. Also in (1), if F is locally finite he proved that every finitely generated nilpotent group is in XF and he conjectured that XF should contain every polycyclic group. This turned out to be very difficult, but a positive solution was eventually found by Roseblade, see (8). Meanwhile Levic in (4) had started a systematic investigation of the classes XF. Although his paper contains a number of errors, obscurities and omissions, it remains an interesting work, and it, or more accurately its recent translation, stimulated this present paper.

Lattices and factorization of polynomials

Abstract: A new algorithm to factorize univariate polynomials over an algebraic number field has been implemented in Algol-68 on a CDC-Cyber 170-750 computer. The algebraic number field is given as the field of rational numbers adjoined by a root of a prescribed minimal polynomial. Unlike other algorithms [1,2] the efficiency of our so-called lattice algorithm does not depend on the irreducibility of the minimal polynomial modulo some prime. The factorization of the polynomial to be factored is constructed from the factorization of that polynomial over a finite field determined by a prime p and an irreducible factor of the minimal polynomial modulo p. The algorithm is based on a theorem on integral lattices and a theorem giving a lower bound for the length of a shortest-length polynomial having modulo pk a non-trivial common divisor with the minimal polynomial. These theorems also enable us to formulate a new algorithm for factoring polynomials over the integers. A technical report describing the algorithms will soon be available from the Mathematisch Centrum, Amsterdam.

On polynomial factorization over finite fields

Abstract: Let f(x) be a polynomial over a finite field F. An algorithm for determining the degrees of the factors of f(x) is presented. As in the Berlekamp algorithm (1968) for determining the factors of f(x), the Frobenius endomorphism on F[x]/(f(x)) plays a central role. Little-known theorems of Schwarz (1956) and Cesaro (1888) provide the basis for the algorithm we present. New and stream-lined proofs of both theorems are provided.

Quadratic modules over polynomial rings

Abstract: One gives the complete solution of the problem of the reduction of quadratic spaces for the polynomial ring over a field.

A unified theory of the point groups and their general irreducible representations

Abstract: Point groups and their general irreducible (vector and projective) representations are characterized by the subgroup conditions for SU(2) in a unified scheme: these conditions are given by simple polynomial equations imposed on the matrix elements of the 2×2 unitary matrices of SU(2). The general irreducible representations for the point groups D∞,Dn (with arbitrary integer n?2), O, and T are given by four simple and effective tables.

Solution of polynomial equations over the complex field

Abstract: Finding all n roots of the general polynomial over the complex field can be difficult theoretically. Therefore, presented here is an approach that attempts to combine algebraic theory (factor theorem) with multi‐stage Monte Carlo optimization to find the n roots of the general polynomial. An example of degree twenty‐five is presented. Rounding error problems for polynomials of degree fifty or over are discussed also.

Formula manipulation in ALGOL 68 and application to Routh's algorithm

Abstract: This paper presents an implementation of Routh's algorithm and the Schur criterion, in order to determine the location of the zeros of a complex polynomial in one variable, over the ring of multivariate polynomials with integral coefficients. Both algorithms are applied to the characteristic polynomials of multistep methods. Moreover, procedures are given for the arithmetic operations +,−,*,÷ with arbitrarily long integral numbers as operands.

Representation of the direct sum of two quadratic fields by rational symmetric matrices

Abstract: Let f be a fourth-degree polynomial over the field of rational numbers Q with leading coefficient 1 which decomposes over Q into the product of two irreducible second-degree polynomials. It is proved that in order that f be the characteristic polynomial of a symmetric matrix with elements in Q, it is necessary and sufficient that all the roots of f be real.

On the additive factorization of homogeneous polynomial matrices

Abstract: An algorithm for implementing para-Hermitian factorization of 2-D homogeneous polynomial matrices is presented. And additive factorization which occurs in extending the factorization of 2-D homogeneous forms to the n-D case, is considered in a class of 2-D scalar polynomials.