On Piecewise Linear Functions and Piecewise Linear Equations

In this paper we treat of the number of solutions of certain piecewise linear (p.l.) equations and of necessary and sufficient conditions for a PL function to be a homeomorphism. The function f: Rⁿ → R^m is PL when a finite set H of hyperplanes exists such that Rⁿ\∪H is the disjoint union of open polyhedral sets C₁, …, C_q and F ∣ C̄ (x) = A_ix + b_i with A_i ∈ R^n,n, B_i ∈ Rⁿ for i = 1, …, q. Letting H* be the set of points common to at least two hyperplanes in ∪ H, we show that when det A₁, … , det A_q all have the same sign then F, restricted to Rⁿ\F₋₁ [F(H*)], is a covering map. From this we conclude that for γ ∈ Rⁿ/F(H*) the number m of solutions of the equation F(x) = γ is independent of γ, whereas for γ ∈ F(H*) it is m at most. The particular case m = 1 provides several alternative sets of necessary and sufficient conditions for F to be a homeomorphism. Sufficient conditions were earlier provided by Fujisawa and Kuh. By one of our theorems equality of the signs of det A₁ …, det A_q is necessary and sufficient for F to be a homeomorphism when at each point of H* the normals to the hyperplanes Hⁱ ∈ H meeting at that point are linearly independent. Theorems for the local homeomorphism of F are provided as well. All the theorems on homeomorphism are also cast in the form of algorithms by which homeomorphism can be determined in practice.