Consider Hardy field $H$, the field of germs of functions at positive infinity.
Can we define $H_I\subset H$, such that it would have the following properties:
$H_I$ is an integral domain.
For each germ $u\in H$ there exists and only one infinitesimally close to it germ $w\in H_I$
All germs of polynomials belong to $H_I$
These properties are not enough to define $H_I$, so here is my question.
Can there be a naturally and simply defined operator on germs that for each germ $u\in H$ would return the infinitesimally close to it germ from $H_I$? For instance the germs $\ln(x)$ and $\ln(x+1)$ are infinitesimally close, so the operator should invariably return only one of these functions (or something else, infinitesimally close to them).
Can we define $H_I$ by some easily verifiable property?