The reference to Stanley's book given by Qiaochu Yuan provides indeed an enlightening exposition on diagonals of bivariate rational functions. Note that the method using a residue mentioned at the end of example 6.3.4 is better suited for computations than the one using Puiseux series (Theorem 6.3.3).

This being said, it is hard (impossible?) to find a reference with a complete algorithm for the computation of algebraic equations for bivariate diagonals in characteristic 0. This is one of the motivations for a recent paper of Alin Bostan, Bruno Salvy and myself on that subject: http://arxiv.org/abs/1510.04526. H.-C. Herbig's request is met there by Algorithm 3 and formula (4).

The reference to Stanley's book given by Qiaochu Yuan provides indeed an enlightening exposition on diagonals of bivariate rational functions. Note that the method using a residue mentioned at the end of example 6.3.4 is better suited for computations than the one using Puiseux series (Theorem 6.3.3).

This being said, it is hard (impossible?) to find a reference with a complete algorithm for the computation of algebraic equations for bivariate diagonals in characteristic 0. This is one of the motivations for a recent paper of Alin Bostan, Bruno Salvy and myself on that subject: Algebraic Diagonals and Walks: Algorithms, Bounds, Complexity. H.-C. Herbig's request is met there by Algorithm 3 and formula (4).