Contraction of Ore Ideals with Applications

Abstract: Ore operators form a common algebraic abstraction of linear ordinary differential and recurrence equations. Given an Ore operator $L$ with polynomial coefficients in $x$, it generates a left ideal $I$ in the Ore algebra over the field $\mathbf{k}(x)$ of rational functions. We present an algorithm for computing a basis of the contraction ideal of $I$ in the Ore algebra over the ring $R[x]$ of polynomials, where $R$ may be either $\mathbf{k}$ or a domain with $\mathbf{k}$ as its fraction field. This algorithm is based on recent work on desingularization for Ore operators by Chen, Jaroschek, Kauers and Singer. Using a basis of the contraction ideal, we compute a completely desingularized operator for $L$ whose leading coefficient not only has minimal degree in $x$ but also has minimal content. Completely desingularized operators have interesting applications such as certifying integer sequences and checking special cases of a conjecture of Krattenthaler.