  We illustrate a step towards the construction of a power counting in
energy-density-functional (EDF) theories, by analyzing the equations of state
(EOSs) of both symmetric and neutron matter. Within the adopted strategy,
next-to-leading order (NLO) EOSs are introduced which contain renormalized
first-order-type terms and an explicit second-order finite part. Employing as a
guide the asymptotic behavior of the introduced renormalized parameters, we
focus our analysis on two aspects: (i) With a minimum number of counterterms
introduced at NLO, we show that each energy contribution entering in the EOS
has a regular evolution with respect to the momentum cutoff (introduced in the
adopted regularization procedure) and is found to converge to a
cutoff-independent curve. The convergence features of each term are related to
its Fermi-momentum dependence. (ii) We find that the asymptotic evolution of
the second-order finite-part coefficients is a strong indication of a
perturbative behavior, which in turns confirms that the adopted strategy is
coherent with a possible underlying power counting in the chosen
Skyrme-inspired EDF framework.
