@article{arreche_2017, title={Computation of the difference-differential Galois group and differential relations among solutions for a second-order linear difference equation}, volume={19}, ISSN={["1793-6683"]}, DOI={10.1142/s0219199716500565}, abstractNote={ We apply the difference-differential Galois theory developed by Hardouin and Singer to compute the differential-algebraic relations among the solutions to a second-order homogeneous linear difference equation [Formula: see text] where the coefficients [Formula: see text] are rational functions in [Formula: see text] with coefficients in [Formula: see text]. We develop algorithms to compute the difference-differential Galois group associated to such an equation, and show how to deduce the differential-algebraic relations among the solutions from the defining equations of the Galois group. }, number={6}, journal={COMMUNICATIONS IN CONTEMPORARY MATHEMATICS}, author={Arreche, Carlos E.}, year={2017}, month={Dec} }
 @article{arreche_singer_2017, title={Galois groups for integrable and projectively integrable linear difference equations}, volume={480}, ISSN={["1090-266X"]}, DOI={10.1016/j.jalgebra.2017.02.032}, abstractNote={We consider first-order linear difference systems over C(x), with respect to a difference operator σ that is either a shift σ:x↦x+1, q-dilation σ:x↦qx with q∈C× not a root of unity, or Mahler operator σ:x↦xq with q∈Z≥2. Such a system is integrable if its solutions also satisfy a linear differential system; it is projectively integrable if it becomes integrable “after moding out by scalars.” We apply recent results of Schäfke and Singer to characterize which groups can occur as Galois groups of integrable or projectively integrable linear difference systems. In particular, such groups must be solvable. Finally, we give hypertranscendence criteria.}, journal={JOURNAL OF ALGEBRA}, author={Arreche, Carlos K. and Singer, Michael F.}, year={2017}, month={Jun}, pages={423–449} }
 @article{arreche_2016, title={On the computation of the parameterized differential Galois group for a second-order linear differential equation with differential parameters}, volume={75}, ISSN={["0747-7171"]}, DOI={10.1016/j.jsc.2015.11.006}, abstractNote={We present algorithms to compute the differential Galois group G associated via the parameterized Picard–Vessiot theory to a parameterized second-order linear differential equation∂2∂x2Y+r1∂∂xY+r0Y=0, where the coefficients r1 and r0 belong to the field of rational functions F(x) over a computable Π-field F of characteristic zero, and the finite set of commuting derivations Π is thought of as consisting of derivations with respect to parameters. This work relies on earlier procedures developed by Dreyfus and by the present author to compute G under the assumption that r1=0, which guarantees that G is unimodular. When r1≠0, we reinterpret a classical change-of-variables procedure in Galois-theoretic terms in order to reduce the computation of G to the computation of an associated unimodular differential Galois group H. We establish a parameterized version of the Kolchin–Ostrowski theorem and apply it to give more direct proofs than those found in the literature of the fact that the required computations can be performed effectively. We then extract from these algorithms a complete set of criteria to decide whether any of the solutions to a parameterized second-order linear differential equation is Π-transcendental over the underlying Π-field of F(x). We give various examples of computation and some applications to differential transcendence.}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, author={Arreche, Carlos E.}, year={2016}, pages={25–55} }