分圆多项式与映射类类型的同调准则
Cyclotomic polynomials and homological criteria for mapping class types
AI总结：
研究闭可定向曲面\(S_g\)上由代数有限型映射类产生的整辛矩阵特征多项式，给出分圆情形下的刻画，即\(n\geq 3\)时，\(\varphi_n(x)\)由代数有限型映射类实现的条件，还得出相关准则及结论。
AI中文摘要：
设\(S_g\)为闭可定向曲面，\(\Psi\colon \mathrm{Mod}(S_g)\to \mathrm{Sp}(2g,\mathbb{Z})\)为由对第一同调群的作用诱导的表示。我们研究由代数有限型映射类产生的整辛矩阵的特征多项式，并在分圆情形下给出完整刻画：对于\(n\geq 3\)，多项式\(\varphi_n(x)\)由代数有限型映射类实现当且仅当\(n\)至多有两个不同素因子。由此可得，若\(n\)无平方因子且至少有三个不同素因子，则每个具有特征多项式\(\varphi_n(x)\)的映射类都是伪阿诺索夫的。这为卡森 - 布莱勒同调准则提供了分圆补充，并给出了仅由伪阿诺索夫映射类实现辛多项式的完整准则。
英文摘要：
Let $S_g$ be a closed orientable surface and let $Ψ\colon \mathrm{Mod}(S_g)\to \mathrm{Sp}(2g,\mathbb{Z})$ be the representation induced by the action on first homology. We investigate the characteristic polynomials of integral symplectic matrices arising from mapping classes of algebraically finite type and give a complete characterization in the cyclotomic case: for $n\geq 3$, the polynomial $φ_n(x)$ is realized by a mapping class of algebraically finite type if and only if $n$ has at most two distinct prime divisors. Consequently, if $n$ is square-free and has at least three distinct prime divisors, then every mapping class with characteristic polynomial $φ_n(x)$ is pseudo-Anosov. This gives a cyclotomic complement to the Casson--Bleiler homological criterion and yields a complete criterion for a symplectic polynomial to be realized only by pseudo-Anosov mapping classes.
