How Many Elements Matter?
  Some studies of stars' multi-element abundance distributions suggest at least
5-7 significant dimensions, but others show that many elemental abundances can
be predicted to high accuracy from [Fe/H] and [Mg/Fe] (or [Fe/H] and age)
alone. We show that both propositions can be, and are, simultaneously true. We
adopt a machine learning technique known as normalizing flow to reconstruct the
probability distribution of Milky Way disk stars in the space of 15 elemental
abundances measured by APOGEE. Conditioning on Teff and log g minimizes the
differential systematics. After further conditioning on [Fe/H] and [Mg/Fe], the
residual scatter for most abundances is $\sigma_{[X/{\rm H}]} \lesssim 0.02$
dex, consistent with APOGEE's reported statistical uncertainties of
$\sim$0.01-0.015 dex and intrinsic scatter of 0.01-0.02 dex. Despite the small
scatter, residual abundances display clear correlations between elements, which
we show are too large to be explained by measurement uncertainties or by the
finite sampling noise. We must condition on at least seven elements to reduce
correlations to a level consistent with observational uncertainties. Our
results demonstrate that cross-element correlations are a much more sensitive
probe of hidden structure than dispersion, and they can be measured precisely
in a large sample even if star-by-star measurement noise is comparable to the
intrinsic scatter. We conclude that many elements have an independent story to
tell, even for the "mundane" disk stars and elements produced by core-collapse
and Type Ia supernovae. The only way to learn these lessons is to measure the
abundances directly, and not merely infer them.
