For each algebraic number $α$ and each positive real number $t$, the $t$-metric Mahler measure $m_t(α)$ 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 {\it exceptional points for $α$}. 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.
The Weil bound and non-exceptional permutation polynomials over finite fields
von zur Gathen的一个著名结果断言，$\mathbb{F}_{q}$上的次数为$n$的非例外置换多项式仅当$q<n^{4}$存在。借助于绝对不可约（可能是奇异的）仿射平面曲线上$\mathbb{F}_{q}$ -点个数的Weil界，Chahal和Ghorpade改进了von zur Gathen的证明，用一个小于$n^{2}(n-2)^{2}$的界来代替$n^{4}$。同样基于Weil界，我们进一步改进了$q$关于$n$的上界，通过一个更简洁和直接的证明。
Permutation polynomials of degree 8 over finite fields of characteristic 2
在线性变换下，我们得到了$\mathbb{F}_{2^r}$上具有$r>3$的次数为$8$的置换多项式的分类.根据[J.数论176（2017）466-66]，次数为$8$ over $\mathbb{F}_{2^r}$的多项式$f$是例外的，当且仅当$f-f(0)$是线性化PP。因此，它足以搜索度$8$在$\mathbb{F}_{2^r}$上的非例外PP，它只存在于$r\leqslant9$由以前的结果。这可能会被运行在个人电脑上的SageMath软件耗尽。为了便于计算，对多项式系数作了线性变换后的要求，并给出了Hermite准则下的显式方程。主要结果是，存在度为$8$ over $\mathbb{F}_{2^r}$（with $r>3$）的非例外PP $f$当且仅当$r\in\{4,5,6\}$，并且这样的$f$被显式列出直到线性变换.
