Bounding the Coefficients of a Divisor of a Given Polynomial.
The coefficients of the greatest common divisor of two polynomials and (GCD) can be obtained from the Sylvester subresultant matrix transformed to lower triangular form, where and deg(GCD) needs to be computed. Firstly, it is supposed that the coefficients of polynomials are given exactly. Transformations of for an arbitrary allowable are in details described and an algorithm for the calculation of the GCD is formulated. If inexact polynomials are given, then an approximate greatest...
The computation of the greatest common divisor (GCD) has many applications in several disciplines including computer graphics, image deblurring problem or computing multiple roots of inexact polynomials. In this paper, Sylvester and Bézout matrices are considered for this purpose. The computation is divided into three stages. A rank revealing method is shortly mentioned in the first one and then the algorithms for calculation of an approximation of GCD are formulated. In the final stage the coefficients...
We develop a recursive method for computing the -removed -orderings and -orderings of order the characteristic sequences associated to these and limits associated to these sequences for subsets of a Dedekind domain This method is applied to compute these objects for and .
In this paper we study an action of the absolute Galois group on bicolored plane trees. In distinction with the similar action provided by the Grothendieck theory of “Dessins d’enfants” the action is induced by the action of on equivalence classes of conservative polynomials which are the simplest examples of postcritically finite rational functions. We establish some basic properties of the action and compare it with the Grothendieck action.
For n ∈ ℕ, L > 0, and p ≥ 1 let be the largest possible value of k for which there is a polynomial P ≢ 0 of the form , , , such that divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that . We find the size of and for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even in that special...
On page 211, line 9, and on page 213, line -6, the assumption should be added that F is not the product of generalized cyclotomic polynomials.