The Levelwise Finite Generation of Free Tambara Functors

We prove the levelwise finite generation of free polynomial $G$-Tambara functors in a collection of cases, most notably when $G$ is a finite Dedekind group or when $G\cong C_p⋊C_q$, $p>q$ primes. In the process, we establish the permanence of various finiteness conditions under box products and norms $n_H^G$ of Tambara functors, including a weak Hilbert Basis Theorem.