Correctness of the global quasi-symmetric quotient algorithm in the general case

Establish whether Algorithm \ref{alg:global}, which selects an ordinary point of two given positive-order operators L,M in C(x)[D], computes a local quasi-symmetric quotient via QuasiSymmetricQuotientAtZero after shifting, and returns it as the global quasi-symmetric quotient, always returns the global quasi-symmetric quotient for arbitrary L and M. Either prove its general correctness or construct a counterexample showing that the algorithm can fail to return the global quasi-symmetric quotient.

Background

The paper introduces symmetric division for linear differential operators with rational function coefficients and defines the global quasi-symmetric quotient as the primitive quotient of maximal order Q such that L ⊗ Q is a right factor of M. Local quasi-symmetric quotients at ordinary points are also defined and linked to the global notion.

Algorithm \ref{alg:global} is proposed: pick an ordinary point ξ common to L and M, shift to x = 0, compute a local quasi-symmetric quotient at zero with Algorithm QuasiSymmetricQuotientAtZero, and shift back to obtain a candidate global quotient. The authors prove that this algorithm is correct in several special regimes (hyperexponential/first-order-factor lclms, C-finite/constant-coefficient operators, and operators with only algebraic solutions).

Empirically, the algorithm consistently succeeds on random instances where M = L ⊗ P with unknown P, suggesting it may work more broadly. However, outside the identified special cases the authors do not have a proof and have not found a counterexample, leaving the general validity unresolved.