Removable Singularities of Ore Operators

Ore algebras are an algebraic structure used to model many different kinds of functional equations like differential and recurrence equations. The elements of an Ore algebra are polynomials for which the multiplication is defined to be usually non-commutative. As a consequence, Gauß’ lemma does not hold in many Ore polynomial rings and hence the product of two primitive Ore polynomials is not necessarily primitive. This observation leads to the distinction of non-removable and removable factors and to the study of desingularizing operators. Desingularization is the problem of finding a left multiple of a given Ore operator in which some factor of the leading coefficient of the original operator is removed. We derive a normal form for such left factors and unify known results for differential and shift operators into one desingularization algorithm. Furthermore, we analyze the effect of removable and non-removable factors on computations with Ore operators. The set of operators of an Ore algebra that give zero when applied to a given function forms a left ideal. The cost of computing an element of this ideal depends on the size of the coefficients (the degree) and the order of the operator. In order to be able to predict or reduce these costs, we derive an order-degree curve. For a given Ore operator, this is a curve in the (r, d)-plane such that for all points (r, d) above this curve, there exists a left multiple of order r and degree d of the given operator. We show how desingularization yields order-degree curves which are extremely accurate in examples. When computed for the generator of an operator ideal from applications like physics or combinatorics, the resulting bound is usually sharp. The generator of a left ideal in an Ore polynomial ring is the greatest common right divisor of the ideal elements, which can be computed by the Euclidean algorithm. Polynomial remainder sequences contain the intermediate results of the Euclidean algorithm when applied to (non-)commutative polynomials. The running time of the algorithm is dependent on the size of the coefficients of the remainders. Different methods have been studied to make these as small as possible. The subresultant sequence of two polynomials is a polynomial remainder sequence in which the size of the coefficients is optimal in the generic case, but when taking the input from applications,