  We associate a quotient of superspace to any hyperplane arrangement by
considering the differential closure of an ideal generated by powers of certain
homogeneous linear forms. This quotient is a superspace analogue of the
external zonotopal algebra, and it further contains the central zonotopal
algebra in the appropriate grading. We show that an evaluation of the bivariate
Tutte polynomial is the bigraded Hilbert series of this quotient. We then use
this fact to construct an explicit basis for the Macaulay inverse. These
results generalize those of Ardila-Postnikov and Holtz-Ron. We also discuss
enumerative consequences of our results in the setting of hyperplane
arrangements.
