The freeness property for simple Danielewski differential algebras

Let [Formula: see text] be a unique factorization domain containing [Formula: see text] and [Formula: see text] be a nonzero non-unit, and let [Formula: see text] be a polynomial which is non-constant modulo every prime factor of [Formula: see text]. Let the [Formula: see text]-algebra [Formula: see text] be endowed with the locally nilpotent [Formula: see text]-derivation [Formula: see text] defined by [Formula: see text] In this paper, we prove that the differential [Formula: see text]-algebra [Formula: see text] has the freeness property if and only if for every prime factor [Formula: see text] of [Formula: see text] the polynomial [Formula: see text], viewed modulo [Formula: see text], is quasi-monic. As a consequence, for every nonzero triangular [Formula: see text]-derivation [Formula: see text] of [Formula: see text] we give a simple and explicit criterion, in terms of [Formula: see text] and [Formula: see text], for [Formula: see text] to have the freeness property.