  We probe the star formation properties of the gas in AzTEC-1 in the COSMOS
field, one of the best resolved and brightest starburst galaxies at $z \approx
4.3$, forming stars at a rate > 1000 $\mathrm{M_{\odot}}\,\mathrm{yr^{-1}}$.
Using recent ALMA observations, we study star formation in the galaxy nucleus
and an off-center star-forming clump and measure a median star formation rate
(SFR) surface density of $\Sigma^{\mathrm{nucleus}}_{\mathrm{SFR}} = 270\pm54$
and $\Sigma^{\mathrm{sfclump}}_{\mathrm{SFR}} =
170\pm38\,\mathrm{M_{\odot}}\,\mathrm{yr}^{-1}\,\mathrm{kpc}^{-2}$,
respectively. Following the analysis by Sharda et al. (2018), we estimate the
molecular gas mass, freefall time and turbulent Mach number in these regions to
predict $\Sigma_{\mathrm{SFR}}$ from three star formation relations in the
literature. The Kennicutt-Schmidt (Kennicutt 1998, KS) relation, which is based
on the gas surface density, underestimates the $\Sigma_{\mathrm{SFR}}$ in these
regions by a factor 2-3. The $\Sigma_{\mathrm{SFR}}$ we calculate from the
single-freefall model of Krumholz et al. 2012 (KDM) is consistent with the
measured $\Sigma_{\mathrm{SFR}}$ in the nucleus and the star-forming clump
within the uncertainties. The turbulence-regulated star formation relation by
Salim et al. 2015 (SFK) agrees slightly better with the observations than the
KDM relation. Our analysis reveals that an interplay between turbulence and
gravity can help sustain high SFRs in high-redshift starbursts. It can also be
extended to other high- and low-redshift galaxies thanks to the high angular
resolution and sensitivity of ALMA observations.
