Entropy and growth rate of periodic points of algebraic Z^d-actions

Expansive algebraic Z^d-actions corresponding to ideals are characterized by the property that the complex variety of the ideal is disjoint from the multiplicative unit torus. For such actions it is known that the limit for the growth rate of periodic points exists and equals the entropy of the action. We extend this result to actions for which the complex variety intersects the multiplicative torus in a finite set. The main technical tool is the use of homoclinic points which decay rapidly enough to be summable.