  We develop a quantitative large deviations theory for random hypergraphs,
which rests on tensor decomposition and counting lemmas under a novel family of
cut-type norms. As our main application, we obtain sharp asymptotics for joint
upper and lower tails of homomorphism counts in the $r$-uniform
Erd\H{o}s--R\'enyi hypergraph for any fixed $r\ge 2$, generalizing and
improving on previous results for the Erd\H{o}s--R\'enyi graph ($r=2$). The
theory is sufficiently quantitative to allow the density of the hypergraph to
vanish at a polynomial rate, and additionally yields tail asymptotics for other
nonlinear functionals, such as induced homomorphism counts.
