On Irregular Binomial $D$-modules

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

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