The absolute value of functions (Q1396770)

Let \(X\) be a topological space and \(f\) be a real valued function defined on a subset of \(X\). We say that \(f\) is absolutely polynomial if there exist continuous functions \(g_0, \dots g_n\) defined on \(X\) such that \( |f(x)|= \sum _{k=0}^{n} g_k(x)f(x)^k \) for all \(x\) in the domain of \(f\). The authors study this notion for \(X = \mathbb R\) and also in more general cases. The paper includes also some applications to the theory of \(\Phi \)-algebras.