Guessing singular dependencies

Given d complex numbers z₁,. . ., z_d, it is classical that linear dependencies λ₁z₁+⋯+λ_dz_d=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 P₁f₁+⋯+P_df_d=0 with P1,...,Pd∈C[z]. Assuming that f₁,. . ., f_d 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 g₁f₁+⋯+g_df_d=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.