Let $A$ be an $\mathcal{O}$-algebra with positive squares and $F\left(
X_{1},...,X_{n}\right) \in\linebreak\mathbb{R}^{+}\left[ X_{1},...,X_{n}\right]
$ be a homogeneous polynomial of degree $p$ $\left( p\in\mathbb{N}^{\ast
},\text{ }p\neq2\right) $. It is shown that for all $0\leq a_{1},...,a_{n}\in
A$ there exists $0\leq a$ $\in A$ such that $F\left( a_{1},...,a_{n}\right)
=a^{p}$ .
As an application we show that every algebra homomorphism $T$ from an
$\mathcal{O}$-algebra $A$\textit{ }with positive squares into an Archimedean
semiprime \textit{f-}algebra $B$ is positive. This improves a result of Render
[14, Theorem 4.1], who proved it for the case of order bounded multiplicative
functional $T$ from an $\mathcal{O}$-algebra $A$ with positive squares into
$\mathbb{R}$.