Restricted Holant Dichotomy on Domains 3 and 4

${\mathrm{Holant}}^{\ast }(f)$ denotes a class of counting problems specified by a constraint function $f$. We prove complexity dichotomy theorems for ${\mathrm{Holant}}^{\ast }(f)$ in two settings: (1) $f$ is any arity-3 real-valued function on input of domain size 3. (2) $f$ is any arity-3 $\{0,1\}$-valued function on input of domain size 4.