(untitled) (Journal of Computational Mathematics, 2021)
A mapping f:Z^n→ R^n is said to possess the direction preserving property if f_i(x)0 implies f_i(y)≥ 0 for any integer points x and y with \|x-y\|_∞≤ 1. In this paper, a simplicial algorithm is developed for computing an integer zero point of a mapping with the direction preserving property. We assume that there is an integer point x^0 with c≤ x^0≤ d satisfying that _1≤ i≤ n(x_i-x^0_i)f_i(x)0 for any integer point x with f(x) eq 0 on the boundary of H={xR^n | c-e≤ x≤ d+e}, where c and d are two finite integer points with c≤ d and e=(1,1,…,1)^R^n. This assumption is implied by one of two conditions for the existence of an integer zero point of a mapping with the preserving property in van der Laan et al. (2004). Under this assumption, starting at x^0, the algorithm follows a finite simplicial path and terminates at an integer zero point of the mapping. This result has applications in general economic equilibrium models with indivisible commodities.