The Geometry of Locally Bounded Rational Functions

This paper develops the geometry of rational functions on non-singular real algebraic varieties that are locally bounded. First various basic geometric and algebraic results regarding these functions are established in any dimension, culminating with a version of {\L}ojasiewicz's inequality. The geometry is further developed for the case of dimension 2, where it can be shown that there exist many of the usual correspondences between the algebra and geometry of these functions that one expects from complex algebraic geometry and from other classes of functions in real algebraic geometry such as regulous functions.