This paper considers the problem of effective algorithms for some problems having structured coefficient matrices. Examples of such problems include rational approximation and rational interpolation. The corresponding coefficient matrices include Hankel, Toeplitz and Vandermonde-like matrices. Effective implies that the algorithms studied are suitable for implementation in either a numeric envi...

We present an algorithm that computes an unmixed-dimensional decomposition of an arbitrary algebraic variety V . Each Vi in the decomposition V = V1 ∪ . . . ∪ Vm is given by a finite set of polynomials which represents the generic points of the irreducible components of Vi. The basic operation in our algorithm is the computation of greatest common divisors of univariate polynomials over extensi...

We start by giving a brief description of the classical Chinese remainder theorem for integers, after whichwe define the greatest common divisor of two polynomials and congruences modulo a polynomial. These concepts allow us to state and prove the Chinese remainder theorem for polynomials. After presenting some important consequences of that theorem, we give its applications to the factorizatio...

We introduce concepts of “recursive polynomial remainder sequence (PRS)” and “recursive subresultant,” along with investigation of their properties. A recursive PRS is defined as, if there exists the GCD (greatest common divisor) of initial polynomials, a sequence of PRSs calculated “recursively” for the GCD and its derivative until a constant is derived, and recursive subresultants are defined...

When people mention the number theoretical achievements in Ancient China, the famous Chinese Remainder Theorem always springs to mind. But, two more of them—the concept of primes and the algorithm for counting the greatest common divisor, are rarely spoken. Some scholars even think that Ancient China has not the concept of primes. The aim of this paper is to show that the concept of primes in A...

Let d be a positive integer. A graph G is called d-divisible if d divides the degree of each vertex of G. G is called nowhere d-divisible if no degree of a vertex of G is divisible by d. For a graph H, gcd(H) denotes the greatest common divisor of the degrees of the vertices of H. The H-packing number of G is the maximum number of pairwise edge disjoint copies of H in G. The H-covering number o...

There exist different approaches to ramification theory of n-dimensional schemes with n ≥ 2. One of these approaches is based on the following idea: to reduce the situation of an n-dimensional scheme X to a number of 1-dimensional settings, just by restricting to curves C properly crossing the ramification subscheme R ⊂ X . This way seems to be very natural; however, it has not got much attenti...