We study how often exceptional configurations of irreducible polynomials over finite fields occur in the context of prime number races and Chebyshev's bias. In particular, we show that three types of biases, which we call “complete bias,” “lower order bias,” and “reversed bias,” occur with probability going to zero among the family of all squarefree monic polynomials of a given degree in as , a power of a fixed prime, goes to infinity. The bounds given extend a previous result of Kowalski, who studied a similar question along particular one-parameter families of reducible polynomials. The tools used are the large sieve for Frobenius developed by Kowalski, an improvement of it due to Perret–Gentil and considerations from the theory of linear recurrence sequences and arithmetic geometry.

中文翻译：

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函数域上素数计数的异常偏差

我们研究在素数竞赛和切比雪夫偏差的背景下，有限域上不可约多项式的异常配置发生的频率。特别是，我们表明，在给定次数的所有无平方一元多项式族中，三种类型的偏差（我们称之为“完全偏差”、“低阶偏差”和“反向偏差”）的发生概率为零。作为，一个固定素数的幂，趋于无穷大。给出的界限扩展了 Kowalski 之前的结果，Kowalski 根据可约多项式的特定单参数族研究了类似的问题。所使用的工具是Kowalski开发的Frobenius大筛，是Perret-Gentil对其的改进，并考虑了线性递推序列和算术几何理论。