On Irregular Binomial $D$-modules

We prove that a holonomic binomial $D$–module $M_A(I,\beta )$ is regular if and only if certain associated primes of $I$ determined by the parameter vector $\beta \in {\text{CC}}^d$ are homogeneous. We further describe the slopes of $M_A(I,\beta )$ along a coordinate subspace in terms of the known slopes of some related hypergeometric $D$–modules that also depend on $\beta$. When the parameter $\beta$ is generic, we also compute the dimension of the generic stalk of the irregularity of $M_A(I,\beta )$ along a coordinate hyperplane and provide some remarks about the construction of its Gevrey solutions.