Simple Shamsuddin derivations of K[X;Y1,…,Yn]K[X;Y1,…,Yn]: An algorithmic characterization

Let KK be a field of characteristic zero, dd a derivation of K[X;Y1,…,Yn]K[X;Y1,…,Yn] of the type d=∂X+∑i=1n(aiYi+bi)∂Yi with ai,bi∈K[X]ai,bi∈K[X] for every ii. We characterize the property “dd is a simple derivation of K[X;Y1,…,Yn]K[X;Y1,…,Yn]” in terms of a certain property of dd, a property that one can effectively check for whether it is satisfied or not. We apply our algorithm to exhibit families of simple derivations of K[X;Y1,…,Yn]K[X;Y1,…,Yn] that are very different from what has been known until now. We also show that our algorithm can be traduced into one that determines effectively whether the solutions in the power series ring K[[t]]K[[t]] of the system of algebraic equations {yi′(t)=ai(t+α)yi(t)+bi(t+α)}i=1n,α∈K, are algebraically independent over K(t)K(t) or not.