  By a numerical continuation method called a diagonal homotopy we can compute
the intersection of two positive dimensional solution sets of polynomial
systems. This paper proposes to use this diagonal homotopy as the key step in a
procedure to intersect general solution sets. Of particular interest is the
special case where one of the sets is defined by a single polynomial equation.
This leads to an algorithm for finding a numerical representation of the
solution set of a system of polynomial equations introducing the equations
one-by-one. Preliminary computational experiments show this approach can
exploit the special structure of a polynomial system, which improves the
performance of the path following algorithms.
