Taming Apparent Singularities via Ore Closure

We consider linear functional equations expressed in the framework of univariate skew polynomial rings. Operations like the removal of apparent singularities of a differential equation, finding the minimal-order recurrence satisfied by the coefficients of its formal solutions, and the desingularization of recurrences and $q$-recurrences share a relation with the calculation of polynomial torsion modules in algebras of skew polynomials. Our analysis here is crucially based on the geometric shape of the singular locus of the functional equation under consideration (fixed singularity, infinite orbit, or periodic orbit). In the differential case, we revisit algorithms by Tsai in view of more efficiency; in the case of recurrences, $q$-recurrences, and Mahler equations, our algorithmic results seem new, and require a more involved machinery.

Joint work with Ph. Dumas, H. Lê, J. Martins, M. Mishna, and B. Salvy.