The finiteness of computing the ultrametric Mahler measure

Recent work of Fili and the author examines an ultrametric version of the Mahler measure, denoted $M_{\mathrm{\infty }}(\alpha )$ for an algebraic number $\alpha$. We show that the computation of $M_{\mathrm{\infty }}(\alpha )$ can be reduced to a certain search through a finite set. Although it is a open problem to record the points of this set in general, we provide some examples where it is reasonable to compute and our result can be used to determine $M_{\mathrm{\infty }}(\alpha )$.