Guessing singular dependencies

Given d complex numbers z1, ..., zd, it is classical that linear dependencies λ1 z1+ ···+ λd zd=0 with λ1, ..., λd∈Z can be guessed using the LLL-algorithm. Similarly, given d formal power series f1, ..., fd ∈ C[[z]], algorithms for computing Padé-Hermite forms provide a way to guess relations P1 f1 + ···+ Pd fd = 0 with P1, ..., Pd ∈C[z]. Assuming that f1, ..., fd have a radius of convergence r > 0 and given a real number R>r, we will describe a new algorithm for guessing linear dependencies of the form g1 f1 + ··· + gd fd = h, where g1, ..., gd, h ∈ C[[z]] have a radius of convergence >R. We will also present two alternative algorithms for the special cases of algebraic and Fuchsian dependencies.