The Geometry of Locally Bounded Rational Functions

TL;DR

This paper explores the geometric properties of locally bounded rational functions on real algebraic varieties, establishing foundational results and extending classical correspondences in the case of surfaces.

Contribution

It develops the geometry of these functions in any dimension and demonstrates classical algebra-geometry correspondences for the two-dimensional case.

Findings

Established basic geometric and algebraic results for locally bounded rational functions.

Proved a version of Łojasiewicz's inequality for these functions.

Showed classical algebra-geometry correspondences hold in the 2D case.

Abstract

This paper develops the geometry of locally bounded rational functions on non-singular real algebraic varieties. 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.