On the Pierce–Birkhoff conjecture in three variables

The so called “Pierce–Birkhoff Conjecture” asserts that a continuous function h on image piecewise polynomial on a finite number of pieces may be written as finitely many Sup and Inf of polynomials. Up to now a positive answer is known for n≤2. In this paper we show that for n=3 such an Inf–Sup description may be obtained outside an arbitrarily small neighborhood of a finite fixed set of points depending only on h.