# Computing Approximate Greatest Common Right Divisors of Differential Polynomials

## TL;DR

This paper introduces an algorithm for computing approximate greatest common right divisors of differential polynomials, extending GCD concepts to non-commutative differential operators with proven local well-posedness and quadratic convergence.

## Contribution

It presents the first algorithm for approximate GCRD of differential polynomials, including a SVD-based initial approximation and convergence analysis.

## Key findings

- Algorithm effectively finds nearby differential polynomials with non-trivial GCRD.
- Proven local well-posedness of the approximate GCRD problem.
- Quadratic convergence of Newton iteration with a good initial guess.

## Abstract

Differential (Ore) type polynomials with "approximate" polynomial coefficients are introduced. These provide an effective notion of approximate differential operators, with a strong algebraic structure. We introduce the approximate Greatest Common Right Divisor Problem (GCRD) of differential polynomials, as a non-commutative generalization of the well-studied approximate GCD problem. Given two differential polynomials, we present an algorithm to find nearby differential polynomials with a non-trivial GCRD, where nearby is defined with respect to a suitable coefficient norm. Intuitively, given two linear differential polynomials as input, the (approximate) GCRD problem corresponds to finding the (approximate) differential polynomial whose solution space is the intersection of the solution spaces of the two inputs. The approximate GCRD problem is proven to be locally well-posed. A method based on the singular value decomposition of a differential Sylvester matrix is developed to produce an initial approximation of the GCRD. With a sufficiently good initial approximation, Newton iteration is shown to converge quadratically to an optimal solution. Finally, sufficient conditions for existence of a solution to the global problem are presented along with examples demonstrating that no solution exists when these conditions are not satisfied.