The Absolute Value of Functions

Abstract

A real-valued function $f$ defined on a topological space is called {\is absolutely polynomial} if its absolute value can be written as a polynomial in $f$ with continuous coefficients. One motivation for studying such functions comes from the theory of rings of continuous functions. While many real functions are absolutely polynomial, we provide a number of interesting explicit examples which are not. The absolutely polynomial criterion turns out to be quite delicate, and we develop the theory in some detail. Our study of absolutely polynomial functions is then widened to more general topological spaces. Our results provide pertinent counterexamples in the theory of rings of quotients of $\Phi$--algebras.