of the terms in the polynomial i.e. how they depend on the external momenta, dictate the form and number of counter-terms needed to be introduced in the Lagrangian in order to cancel the UV divergences of the theory. E.g. in the above case, they tell us that we need two counter-terms to renormalize the two-point function and they shall be of the form $- \frac{A}{2} (\partial \phi)^2 $ and $- \frac{B}{2} m^2 \phi^2$. In fact, this is how we get convinced that the correlators can be renormalized at all.

4.  In contrast, for a logarithmically divergent integral $$I_L = \int \frac{d^4 \ell}{(\ell^2 + \Delta)^2 } \; ,$$ following the above procedure shall simply give $$\label{eq:log_usual}
        I_L = \left( \frac{F_{-1}}{\epsilon} + \frac{F_{-1}}{2} \log \left( \frac{\mu^2}{\Delta} \right) + F_0 + \cdots \right) \; .$$ It is clear that for a logarithmically divergent integral, the dependence on external momenta does not turn up. We could say that for this case, the polynomial is just one. This causes the counterterm to be just a factor such as $Z_g g$ (where $g$ is the coupling constant and $Z_g$ is the renormalizing $Z$ factor).

5.  The function ${\cal F}$ has this property that in the limit 