On Solutions of Linear Ordinary Difference Equations in their Coefficient Field

We describe a rational algorithm for nding the denominator of any solution of a linear ordinary diierential equation in its coeecient eld. As a consequence, there is now a rational algorithm for nding all such solutions when the coeecients can be built up from the rational functions by nitely many algebraic and primitive adjunctions. This also eliminates one of the computational bottlenecks in algorithms that either factor or search for Liouvillian solutions of such equations with Liouvillian coeecients.