On the Pierce-Birkhoff conjecture in three variables

The so called ”Pierce-Birkhoff Conjecture” asserts that a continuous function h on Rn 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. The problem known as ”Pierce-Birkhoff Conjecture” (PBC in short) finds its origin in a paper by G. Birkhoff and R. S. Pierce [BP]. The question is the following. Let h be a continuous piecewise polynomial function on R, with a finite number of pieces. Is it possible to describe h starting from polynomial functions and using finitely many Sup and Inf operations ? It is not hard to show (see [HI] for explicit formulas) that the set ISD(n) of such Inf-Sup-Definable functions on R is a ring and thus a subring of the ring of continuous piecewise polynomial functions on R (say PWP(n)). The question is about the other inclusion. It can be mentionned that this problem presents several analogies with more classical questions as Hilbert Nullstellensatz, real Nullstellensatz or Positivstellensatz (i. e. Hilbert’s 17th problem). For example, we may think that among piecewise polynomial functions, the continuity is the geometric condition (analogous to the positivity in the latter case) and being defined by Sup and Inf is the algebraic condition (analogous to be defined as sum of squares). A positive answer to the question for n = 2 has been given in 1983 by the author [Mah1] and a proof has been sketched in a short note [Mah2]. As