  When atoms or molecules are exposed to strong short-pulse infrared radiation,
ionization as well as "frustrated tunneling ionization" (FTI) can occur, in
which a portion of the almost ionized electrons recombine into the initial
ground or an excited bound state. We analyze the ellipticity dependence of the
relative signals that are predicted in a single-active electron approximation
(SAE), the validity of which is checked against a parameter-free multi-electron
\hbox{$R$-matrix} (close-coupling) with time dependence approach. We find good
agreement between the results from both models, thereby providing confidence in
the SAE model potential to treat the process of interest. Comparison of the
relative excitation probabilities found in our numerical calculations with the
predictions of Landsman {\it et al.}\ (New Journal of Physics {\bf 15} (2013)
013001) and Zhao {\it et al.}\ (Optics Express {\bf 27} (2019) 21689) reveals
good agreement with the former for short pulses. For longer pulses, the
ellipticity dependence becomes wider than that obtained from the Landsman {\it
et al.} formula, but we do not obtain the increase compared to linearly
polarized radiation predicted by Zhao {\it et al.}
