On Piecewise Linear Functions and Piecewise Linear Equations

Abstract
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 → Rm is PL when a finite set H of hyperplanes exists such that Rn\∪H is the disjoint union of open polyhedral sets C1, …, Cq and F ∣ C̄ (x) = Aix + bi with Ai ∈ Rn,n, Bi ∈ 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 Aq 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 Aq 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.

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