The Pierce-Birkhoff Conjecture
by
James J. Madden
Louisiana State University

The Pierce-Birkhoff Conjecture (PBC) dates from about 1960. The most common formulation is due to John Isbell: ``Is every real-valued continuous piecewise polynomial function on real affine n-space expressible using finitely many polynomial functions and the operations of (pointwise) supremum and infimum?'' The answer is ``yes'' for n=1, 2, (Mahé, 1983), but is unknown for n >= 3. A solution for n >= 3 seems to require a better understanding than we currently have of the relationship between certain algebraic and topological features of real singularities.

The real spectrum enables one to formulate a version of the PBC that makes sense for an arbitrary ring. The classical theory of Enriques and Zariski on complete ideals in 2-dimensional regular local rings provides the tools needed to extend Mahé's 1983 result to arbitrary 2-dimensional regular real algebras. With a generalization of some specific parts of the Enriques-Zariski theory to 3 dimensions, one would have a proof of the PBC for n=3. We shall describe the generalization needed and present some interesting examples related to sequences of point blow-ups along a valuation centered on a 3-dimensional regular local ring which show some of the obstacles to proving the needed generalization.

Date received: July 5, 1999