3.3. The greatest common divisor and the least common multiple 12 3.4. Parabolic divisors 14 3.5. Divisibility: further properties 14 4. Normal forms arising from HNN extensions 16 4.1. HNN -normal form 16 5. Conjugacy problem 18 5.1. Conjugacy Criterion for HNN -extensions 18 5.2. Block decomposition 22 5.3. Conjugacy with respect to a subgroup 23 5.4. An algorithm for solving the Conjugacy Pr...

We consider the following computational problem: we are given two coprime univariate polynomials f0 and f1 over a ring R and want to find whether after a small perturbation we can achieve a large gcd. We solve this problem in polynomial time for two notions of “large” (and “small”): large degree (when R = F is an arbitrary field, in the generic case when f0 and f1 have a so-called normal degree...

This paper presents a regularization theory for numerical computation of polynomial greatest common divisors and a convergence analysis, along with a detailed description of a blackbox-type algorithm. The root of the ill-posedness in conventional GCD computation is identified by its geometry where polynomials form differentiable manifolds entangled in a stratification structure. With a proper r...

We consider the problem of finding for a given N -tuple of polynomials (real or complex) the closest N -tuple that has a common divisor of degree at least d. Extended weighted Euclidean seminorm of the coefficients is used as a measure of closeness. Two equivalent representations of the problem are considered: (i) direct parameterization over the common divisors and quotients (image representat...

This paper about Greater Common Divisor GCD, the paper shows that there is a lot of algorithms, some of these algorithm is good in timing and make low number of iteration, the other make a lot of iteration with a lot of time! But as we see in the analysis of the algorithms that some of the algorithms is faster than the others in small numbers (like Brute force is faster than Bishop Algorithm in...

The extended greatest common divisor GCD problem is given a vector a a an of positive integers compute g the greatest common divisor of these inte gers and nd a vector x x xn of integer coe cients such that