  This paper contains a partial result on the Pierce--Birkhoff conjecture on
piece-wise polynomial functions defined by a finite collection {f 1,. .., f r}
of polynomials. In the nineteen eighties, generalizing the problem from the
polynomial ring to an artibtrary ring $\Sigma$, J. Madden proved that the
Pierce--Birkhoff conjecture for $\Sigma$ is equivalent to a statement about an
arbitrary pair of points $\alpha$, $\beta$ $\in$ Sper $\Sigma$ and their
separating ideal < $\alpha$, $\beta$ >, we refer to this statement as the local
Pierce-Birkhoff conjecture at $\alpha$, $\beta$. In [8] we introduced a
slightly stronger conjecture, also stated for a pair of points $\alpha$,
$\beta$ $\in$ Sper $\Sigma$ and the separating ideal < $\alpha$, $\beta$ >,
called the Connectedness conjecture, about a finite collection of elements {f
1, . . ., fr} $\subset$ $\Sigma$. In the paper [10] we introduced a new
conjecture, called the Strong Connectednessconjecture, and proved that the
Strong Connectedness conjecture in dimension n--1 implies the Strong
Connectedness conjecture in dimension n in the case when ht(< $\alpha$, $\beta$
>) $\le$ n -- 1.The Pierce-Birkhoff Conjecture for r = 2 is equivalent to the
Connectedness Conjecture for r = 1, this conjecture is called the Separation
Conjecture. The Strong Connectedness Conjecture for r = 1 is called the Strong
Separation Conjecture. In the present paper, we fix a polynomial f $\in$ R[x,
z] where R is a real closed field and x = (x1, . . ., xn), z are n + 1
independent variables. We define the notion of two points $\alpha$, $\beta$
$\in$ Sper R[x, z] being in good position with respect to f. The main result of
this paper is a proof of the Strong Separation Conjecture in the case when
$\alpha$ and $\beta$ are in good position with respect to f.

Algebraic Geometry