Polynomial and rational solutions of holonomic systems

In this article, we give two new algorithms to find the polynomial and rational function solutions of a given holonomic system associated to a set of linear differential operators in the Weyl algebra D=kleft angle bracketx1,…,xn,∂1,…,∂nright-pointing angle bracket, where k is a computable subfield of the complex numbers. Both algorithms are based on the theory of D-modules – the first algorithm obtains degree bounds on the solutions through Gröbner deformations and b-functions while the second algorithm evaluates the dimension of the solutions through duality and restriction.