Utility functions for Debreu's ‘excess demands’

Given an arbitrary function x: $\text{R}$^l→$\text{R}$^l satisfying Walras law and homogeneity, Debreu decomposed x into the sum of l ‘individually rational’ functions x(p)=Σ^l_k=1[uvbar|x]^k(p). Here we find explicit utility functions u^k, constructed on the basis of a simple geometric intuition, which give rise to Debreu's excess demands [uvbar|x]^k(p).