  The origin of the high SFR observed in high-z dusty star-forming galaxies is
still unknown. Large fractions of dense molecular gas might provide part of the
explanation, but there are few observational constraints on the amount of dense
gas in high-z systems dominated by star formation. We present the results of
our ALMA program targeting dense-gas tracers (HCN(5-4), HCO+(5-4), and
HNC(5-4)) in 5 strongly lensed galaxies from the SPT SMG sample. We detected
two of these lines (SNR>5) in SPT-125-47 at z=2.51 and tentatively detected all
three (SNR~3) in SPT0551-50 at z=3.16. Since a significant fraction of our
target lines is not detected, we developed a statistical method to derive
unbiased mean properties taking into account both detections and
non-detections. On average, the HCN(5-4) and HCO+(5-4) luminosities of our
sources are a factor of ~1.7 fainter than expected, based on the local
L'HCN(5-4)-LIR relation, but this offset corresponds to only ~2 sigma. We find
that both the HCO+/HCN and HNC/HCN flux ratios are compatible with unity. The
first ratio is expected for PDRs while the second is consistent with PDRs or
XDRs and/or mid-IR pumping of HNC. Our sources are at the high end of the local
relation between the star formation efficiency, determined using the LIR/[CI]
and LIR/CO ratios, and the dense gas fraction, estimated using the HCN/[CI] and
HCN/CO ratios. In SPT0125-47, we found that the velocity profiles of the lines
tracing dense (HCN, HCO+) and lower-density (CO, [CI]) gas are similar. In
addition to these lines, we obtained one robust and one tentative detection of
13CO(4-3) and found an average I12CO(4-3)/I13CO(4-3) flux ratio of
26.1$_{-3.5}^{+4.5}$, indicating a young but not pristine interstellar medium.
We argue that the combination of large and slightly enriched gas reservoirs and
high dense-gas fractions could explain the prodigious star formation in these
systems.
