Solving discrete zero point problems

In this paper we present two theorems on the existence of a discrete zero point of a function from the n-dimensional integer lattice Z(n) to the n-dimensional Euclidean space R-n. The theorems differ in their boundary conditions. For both theorems we give a proof using a combinatorial lemma and present a constructive proof based on a simplicial algorithm that finds a discrete zero point within a finite number of steps.