  Cyclic algebraic Z^d-actions are defined by ideals of Laurent polynomials in
d commuting variables. Such an action is expansive precisely when the complex
variety of the ideal is disjoint from the multiplicative d-torus. For such
expansive actions it is known that the limit for the growth rate of periodic
points exists and is equal to the entropy of the action. In an earlier paper
the authors extended this result to ideals whose variety intersects the d-torus
in a finite set. Here we further extend it to the case when the dimension of
intersection of the variety with the d-torus is at most d-2. The main tool is
the construction of homoclinic points which decay rapidly enough to be
summable.
