  For each algebraic number $\alpha$ and each positive real number $t$, the
$t$-metric Mahler measure $m_t(\alpha)$ creates an extremal problem whose
solution varies depending on the value of $t$. The second author studied the
points $t$ at which the solution changes, called {\it exceptional points for
$\alpha$}. Although each algebraic number has only finitely many exceptional
points, it is conjectured that, for every $N \in \mathbb N$, there exists a
number having at least $N$ exceptional points. In this article, we describe a
polynomial time algorithm for establishing the existence of numbers with at
least $N$ exceptional points. Our work constitutes an improvement over the best
known existing algorithm which requires exponential time. We apply our main
result to show that there exist numbers with at least $37$ exceptional points,
another improvement over previous work which was only able to reach $11$
exceptional points.
