Unlike the ordinary supernovae (SNe) some of which are hydrogen and helium deficient (called Type Ic SNe), broad-lined Type Ic SNe (SNe Ic-bl) are very energetic events, and all SNe coincident with bona fide long duration gamma-ray bursts (LGRBs) are of Type Ic-bl. Understanding the progenitors and the mechanism driving SN Ic-bl explosions vs those of their SNe Ic cousins is key to understanding the SN-GRB relationship and jet production in massive stars. Here we present the largest set of host-galaxy spectra of 28 SNe Ic and 14 SN Ic-bl, all discovered before 2013 by the same untargeted survey, namely the Palomar Transient Factory (PTF). We carefully measure their gas-phase metallicities, stellar masses (M*s) and star-formation rates (SFRs) by taking into account recent progress in the metallicity field and propagating uncertainties correctly. We further re-analyze the hosts of 10 literature SN-GRBs using the same methods and compare them to our PTF SN hosts with the goal of constraining their progenitors from their local environments by conducting a thorough statistical comparison, including upper limits. We find that the metallicities, SFRs and M*s of our PTF SN Ic-bl hosts are statistically comparable to those of SN-GRBs, but significantly lower than those of the PTF SNe Ic. The mass-metallicity relations as defined by the SNe Ic-bl and SN-GRBs are not significantly different from the same relations as defined by the SDSS galaxies, in contrast to claims by earlier works. Our findings point towards low metallicity as a crucial ingredient for SN Ic-bl and SN-GRB production since we are able to break the degeneracy between high SFR and low metallicity. We suggest that the PTF SNe Ic-bl may have produced jets that were choked inside the star or were able break out of the star as unseen low-luminosity or off-axis GRBs.
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