The description of generalized translation-invariant $p$-adic Gibbs measures for the Potts model on the Cayley tree of order three

We describe generalized translation-invariant $p$-adic Gibbs measures for the Potts model on the Cayley tree of order three, confirming and extending the related results in the literature. We introduce a cube root function over the $p$-adic field $\mathbb{Q}_p$, which enables us to explicitly prescribe all roots of the cubic monomial equation over $\mathbb{Q}_p$.