  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.
