Interpreting the scaling relations followed by galaxies is a fundamental toolfor assessing how well we understand galaxy formation and evolution. Severalscaling relations involving the galaxy metallicity have been discovered throughthe years, the foremost of which is the scaling with stellar mass. Thisso-called mass-metallicity relation is thought to be fundamental and has beensubject to many studies in the literature. We study the dependence of thegas-phase metallicity on many different galaxy properties to assess which ofthem determines the metallicity of a galaxy. We applied a random forestregressor algorithm on a sample of more than 3000 nearby galaxies from theSDSS-IV MaNGA survey. Using this machine-learning technique, we explored theeffect of 148 parameters on the global oxygen abundance as an indicator of thegas metallicity. $M_{\rm \star}$/$R_e$, as a proxy for the baryonicgravitational potential of the galaxy, is found to be the primary factordetermining the average gas-phase metallicity of the galaxy ($Z_g$). Itoutweighs stellar mass. A subsequent analysis provides the strongest dependenceof $Z_g$ on $M_\star / R_e^{\,0.6}$. We argue that this parameter traces thetotal gravitational potential, and the exponent $\alpha\simeq 0.6$ accounts forthe inclusion of the dark matter component. Our results reveal the importanceof the relation between the total gravitational potential of the galaxy and thegas metallicity. This relation is tighter and likely more primordial than thewidely known mass-metallicity relation.

Context. One of the most fundamental scaling relations in galaxies isobserved between metallicity and stellar mass -- the mass-metallicity relation(MZR) -- although recently a stronger dependence of the gas-phase metallicitywith the galactic gravitational potential ($\Phi \rm ZR$) has been reported.Further dependences of metallicity on other galaxy properties have beenrevealed, with the star formation rate (SFR) being one of the most studied anddebated secondary parameters in the relation (the so-called fundamentalmetallicity relation). Aims. In this work we explore the dependence of thegas-phase metallicity residuals from the MZR and $\Phi \rm ZR$ on differentgalaxy properties in the search for the most fundamental scaling relation ingalaxies. Methods. We applied a random forest regressor algorithm on a sampleof 3430 nearby star-forming galaxies from the SDSS-IV MaNGA survey. Using thistechnique, we explored the effect of 147 additional parameters on the globaloxygen abundance residuals obtained after subtracting the MZR. Alternatively,we followed a similar approach with the metallicity residuals from the $\Phi\rm ZR$. Results. The stellar metallicity of the galaxy is revealed as thesecondary parameter in both the MZR and the $\Phi \rm ZR$, ahead of the SFR.This parameter reduces the scatter in the relations $\sim 10-15\%$. We find the3D relation between gravitational potential, gas metallicity, and stellarmetallicity to be the most fundamental metallicity relation observed ingalaxies.