Restricted Holant Dichotomy on Domains 3 and 4

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