Abstract: A function f : Rⁿ -> R is said to be piecewise polynomial if there exist finitely many polynomials f_i in n variables such that for every point a in Rⁿ we have f(a) = f_i(a) for at least one f_i. The celebrated Pierce-Birkhoff conjecture asserts that every piecewise polynomial function f on Rⁿ can be obtained from a finite collection of polynomials by iterating the operations of maximum and minimum. This is equivalent to saying that there exists a finite collection f_{ij} of polynomials such that f = max limits_i (min limits_j f_{ij}). In this lecture, I will describe an approach to proving this conjecture proposed by J. Madden in the nineteen eighties and which we continue to  develop more recently with F. Lucas, D. Schaub and J. Madden. Our key tool is  the real spectrum of a ring; a large part of the lecture will be devoted to introducing the real spectrum.