Real Algebraic and Analytic Geometry |

On the Pierce-Birkhoff Conjecture in three variables.

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Submission: 2006, February 17.

*Abstract:
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.

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