The fundamental metallicity relation emerges from the local
  anti-correlation between star formation rate and gas-phase metallicity
  existing in disk galaxies
  The fundamental metallicity relation (FMR) states that galaxies of the same
stellar mass but larger star formation rate (SFR) tend to have smaller
gas-phase metallicity (<Zg>). It is thought to be fundamental because it
naturally arises from the stochastic feeding of star-formation from external
metal-poor gas accretion, a process extremely elusive to observe but essential
according the cosmological simulations of galaxy formation. In this letter, we
show how the FMR emerges from the local anti-correlation between SFR surface
density and Zg recently observed to exist in disk galaxies. We analytically
derive the global FMR from the local law, and then show that both relations
agree quantitatively when considering the star-forming galaxies of the MaNGA
survey. Thus, understanding the FMR becomes equivalent to understanding the
origin of the anti-correlation between SFR and metallicity followed by the set
of star-forming regions of any typical galaxy. The correspondence between local
and global laws is not specific of the FMR, so that a number of local relations
should exist associated with known global relations.
