Polynomial and Rational Functions

One of the unifying themes of modern geometry is that in order to understand any class of geometric objects, one should also study functions on those objects and mappings between those objects, especially the mappings which preserve some geometric property of interest. For instance, in topology one considers the class of continuous functions on topological spaces; in differential geometry one considers the class of infinitely differentiable functions (i.e., smooth functions) on manifolds (smooth mappings between manifolds); in (real) complex analytic geometry one considers the class of (real) complex analytic functions.

Question To study the geometry of an algebraic set, for example, the singularity of points, what class of functions can we use?

Surprisingly simple, algebraic geometry only uses polynomial functions (plus rational functions which are quotients of polynomial functions) on algebraic sets and polynomial mappings (plus rational mappings) between algebraic sets. If we take an open set U in the com34plex metric space Cⁿ and write https://www.niso.org/standards/z39-96/ns/oasis-exchange/table"> C°(U) for the ring of all continuous functions, C^∞(U) for the ring of all smooth functions, C^ω(U) for the ring of all analytic functions, C[x] for the ring of all polynomial functions, where C[x] = C[x₁,...,x_n], then we have the following strict inclusion relations

https://www.w3.org/1998/Math/MathML">IR[x]⊂Cω(U)⊂C∞(U)⊂C∘(U).https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315274515/64311f82-81f6-4c52-89ea-d39dfd59d7fa/content/inline-math83.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

In this chapter, we introduce the polynomial mappings between algebraic sets and start the study of polynomial functions and rational functions on algebraic sets. The results of this investigation will constitute another chapter of the “algebra-geometry dictionary” that we started in Chapter I. The algebraic properties of polynomial and rational functions on an algebraic set yield insight into the geometric properties of the algebraic set itself. So this chapter motivates the “local” study of algebraic sets in CH.VI.