Guessing singular dependencies

Given d   complex numbers z1,…,zdz1,…,zd, it is classical that linear dependencies λ1z1+⋯+λdzd=0λ1z1+⋯+λdzd=0 with λ1,…,λd∈Zλ1,…,λd∈Z can be guessed using the LLL-algorithm. Similarly, given d   formal power series f1,…,fd∈C[[z]]f1,…,fd∈C[[z]], algorithms for computing Padé–Hermite forms provide a way to guess relations P1f1+⋯+Pdfd=0P1f1+⋯+Pdfd=0 with P1,…,Pd∈C[z]P1,…,Pd∈C[z]. Assuming that f1,…,fdf1,…,fd have a radius of convergence r>0r>0 and given a real number R>rR>r, we will describe a new algorithm for guessing linear dependencies of the form g1f1+⋯+gdfd=hg1f1+⋯+gdfd=h, where g1,…,gd,h∈C[[z]]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.