Divisors Of Polynomials Research Articles

A note on abnormal polynomial remainder sequences

Polynomial Remainder Sequences (PRS) have been used in the computation of greatest common divisors of polynomials for many years. We call a polynomial division abnormal if the degree of the remainder is strictly less than the degree of the divisor minus one; an abnormal PRS is one with at least one abnormal division. An abnormal polynomial division in general implies a nontrivial greatest common divisor among the coefficients of the remainder; this can lead to inefficiencies in the later computations in the PRS.These results suggest that the "classical" PRS methods may have not exploited all of the algebraic structure available in the problem.

Part I-Smith form and common divisor of polynomial matrices

Polynomial matrices play an important part in linear system calculations. New computational procedures are given for calculation of the Smith normal form and the greatest common right divisor of polynomial matrices. It is shown how suitable transformation matrices can be determined for the calculation of the Smith normal form, and how a set of polynomial matrix multipliers can be calculated for the greatest common right divisor problem. Neither of these algorithms relies on explicit calculation of the greate3t common divisor of polynomials. Limited numerical experience has shown that the3e algorithms are both fast and accurate.

Greatest common divisor of several polynomials

AbstractGiven a polynomial a(λ) with degree n, and polynomials b1(λ), …, bm(λ) of degree not greater than n – 1, then the degree k of the greatest common divisor of the polynomials is equal to the rank defect of the matrix R = [b1(A), b2(A), …, bm(A)], where A is a suitable companion matrix of a(λ). Furthermore, it is shown that if the first k rows of R are expressed as linear combinations of the remaining n – k rows (which are linearly independent) then the greatest common divisor is given by the coefficients of row k + 1 in these expressions. A simple expression is derived for R and a permutation of the columns of this matrix establishes a direct connexion with controllability of a constant linear control system. Finally, when m = 1 a relationship between the corresponding R and Sylvester's matrix is exhibited.

On the Prime Divisors of Polynomials

On the Prime Divisors of Polynomials

On the Prime Divisors of Polynomials

(1971). On the Prime Divisors of Polynomials. The American Mathematical Monthly: Vol. 78, No. 3, pp. 250-266.

Divisors of polynomials and power series with positive coefficients

Divisors of polynomials and power series with positive coefficients