Greatest common divisor

The Fourier Transform of Functions of the Greatest Common Divisor

We study discrete Fourier transformations of functions of the greatest common divisor: n ∑ k=1 f((k, n)) · exp( − 2πikm/n). Euler’s totient function: φ(n) = n ∑ k=1 (k, n) · exp(−2πik/n) is an example. The greatest common divisor (k, n) = n ∑ m=1 exp(2πikm/n) · ∑ d|n cd(m) d is another result involving Ramanujan’s sum cd(m). The last equation, interestingly, can be evaluated for k in the comple...

Barnett's Theorems About the Greatest Common Divisor of Several Univariate Polynomials Through Bezout-like Matrices

This article provides a new presentation of Barnett’s theorems giving the degree (resp. coefficients) of the greatest common divisor of several univariate polynomials with coefficients in an integral domain by means of the rank (resp. linear dependencies of the columns) of several Bezout-like matrices. This new presentation uses Bezout or hybrid Bezout matrices instead of polynomials evaluated ...

Divisibility Structure and Finitely Generated Ideals in the Disc Algebra

The main purpose of this paper is the following algebraic generalization of the corona theorem for the disc algebra A (D): I f d is a greatest common divisor of the functions f l . . . . . fn cA (D), then there exist functions gl,.",gnEA (D) with d-~flgl q-... q-fngn. This generalization is false for many algebras of holomorphic functions, e. g. in case of the Banach algebra H ~ Under the assum...

PartII Number Theory

1 Division 3 1.1 Division Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Greatest common divisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Euclidean Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Fundamental theorem of arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . ....

The calculation of the degree of an approximate greatest common divisor of two polynomials