In this paper, we are concerned with the problem of counting the
multiplicities of a zero-dimensional regular set's zeros. We generalize
the squarefree decomposition of univariate polynomials to the so-called
pseudo squarefree decomposition of multivariate polynomials, and then
propose an algorithm for decomposing a regular set into a finite number
of simple sets. From the output of this algorithm, the multiplicities of
zeros could be directly read out, and the real solution isolation with
multiplicity can also be easily produced. As a main theoretical result
of this paper, we analyze the structure of dual space of the saturated
ideal generated by a simple set as well as a regular set. Experiments
with a preliminary implementation show the efficiency of our method.