Real Algebraic and Analytic Geometry
Submission: 2006, February 17.
The problem known as "Pierce-Birkhoff Conjecture" is the following. Let h be a continuous piecewise polynomial function in n variables, with a finite number of pieces. Is it possible to describe h starting from polynomial functions and using a finite number of Sup and Inf operations ? As far as we know, no result is known in more than two variables. In this paper we present a new result for n=3, namely
Theorem 1. Given such a function h in 3 variables, there exists a finite number of points such that h is Inf-Sup Definable outside a union of balls of arbitrarily small radius centered at the points.
Theorem 2. Given such a function h in 3 variables, there exist a polynomial g with finitely many poles such that gh is ISD .
Mathematics Subject Classification (2000): 14P10,06A11.
Full text, 20p.: dvi 90k, ps.gz 164k, pdf 207k.