Piecewise Linearization of Real Analytic Functions

Many mathematicians considered the problem of triangulations of algebraic sets, analytic sets, semi-algebraic sets, etc. [2], [4], [7], [8], [9]. [10], [14] and [19]. We want to consider more generally a global piecewise linearization of real analytic functions. For this, we need simultaneous triangulations of all levels of the functions. A function on a polyhedron is called piecewise linear (=PL, if we have a triangulation of the polyhedron such that the restriction of the function on each simplex is linear. A C°° triangulation of a real analytic manifold M is a pair of a simplicial complex K and a homeomorphism g : \K\—*My ,A"| meaning the underlying polyhedron of K, such that the restriction of g on each simplex is a C°° diffeomorphism onto the image. The existence of C°° triangulation is wellknown (e.g. [12]). In this paper manifolds have not boundary unless otherwise specified.