  This paper is a step in our program for proving the Piece-Birkhoff Conjecture
for regular rings of any dimension (this would contain, in particular, the
classical Pierce-Birkhoff conjecture which deals with polynomial rings over a
real closed field). We first recall the Connectedness and the Definable
Connectedness conjectures, both of which imply the Pierce - Birkhoff
conjecture. Then we introduce the notion of a system of approximate roots of a
valuation v on a ring A (that is, a collection Q of elements of A such that
every v-ideal is generated by products of elements of Q). We use approximate
roots to give explicit formulae for sets in the real spectrum of A which we
strongly believe to satisfy the conclusion of the Definable Connectedness
conjecture. We prove this claim in the special case of dimension 2. This proves
the Pierce-Birkhoff conjecture for arbitrary regular 2-dimensional rings.
