For each algebraic number α and each positive real number t, the t-metric Mahler measure mt(α) 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 exceptional points forα . Although each algebraic number has only finitely many exceptional points, it is conjectured that, for every N ∈ ℕ, there exists a number having at least N exceptional points. In this paper, 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.

中文翻译：

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用于检测具有许多异常点的数字的多项式时间测试

对于每个代数数α和每个正实数吨， 这吨公制马勒测度米吨(α)创建一个极值问题，其解决方案取决于吨. 第二作者研究点吨解决方案发生变化的位置，称为特殊点α . 尽管每个代数数只有有限多个例外点，但可以推测，对于每个ñ ∈ ℕ, 存在一个数至少有ñ异常点。在本文中，我们描述了一种多项式时间算法，用于建立至少有ñ异常点。我们的工作构成了对需要指数时间的最知名的现有算法的改进。我们应用我们的主要结果来表明存在至少37特殊点，对以前只能达到的工作的另一个改进11异常点。