  Coprimeness property was introduced to study the singularity structure of
discrete dynamical systems. In this paper we shall extend the coprimeness
property and the Laurent property to further investigate discrete equations
with complicated pattern of singularities. As examples we study extensions to
the Somos-$4$ recurrence and the two-dimensional discrete Toda equation. By
considering their non-autonomous polynomial forms, we prove that their tau
function analogues possess the extended Laurent property with respect to their
initial variables and some extra factors related to the non-autonomous terms.
Using this Laurent property, we prove that these equations satisfy the extended
coprimeness property. This coprimeness property reflects the singularities that
trivially arise from the equations.
