Real Algebraic and Analytic Geometry

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137. Louis Mahé:
On the Pierce-Birkhoff Conjecture in three variables.


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.

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