A Polynomial Time Test to Detect Number with Many Exceptional Points

TL;DR

This paper introduces a polynomial time algorithm to identify algebraic numbers with many exceptional points related to the $t$-metric Mahler measure, significantly improving efficiency over previous methods and demonstrating numbers with at least 37 exceptional points.

Contribution

The authors develop a polynomial time algorithm to detect algebraic numbers with numerous exceptional points, advancing beyond the exponential time algorithms previously used.

Findings

Algorithm confirms existence of numbers with at least 37 exceptional points

Improves computational efficiency from exponential to polynomial time

Demonstrates the practical application of the algorithm to specific cases

Abstract

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.