AI 中文总结
研究根系zeta函数，证明特定超平面是极除子，推导递归旗标留数等公式，恢复经典奇异数据及相关留数函数，确定$B_3$和$C_3$的一些性质，如双极点位置等。
AI 中文摘要
设$\Phi$为不可约晶体根系，$Z_\Phi(\mathbf{s})$为$\Phi$的非扭曲小森 - 松本 - 津村zeta函数，每个正余根对应一个复指数；其对角特化是单变量维滕zeta函数。对于简单节点的非空集$S$和整数$\ell\geq0$，令$H_{S,\ell}$为根与$S$相交的指数之和为$|S| - \ell$的超平面。我们证明每个适当支撑超平面$H_{S,\ell}$在一般点是真正的极除子，而精确齐次性仅留下未平移的全支撑除子；每个一般留数是射影周期和互补子系统的多项式加权zeta函数的显式有限泰勒 - 喷流和。在最大支撑美妙模型上，边界贡献由严格装饰旗标索引。我们推导递归旗标留数、分量质量伽马因子以及任何横向切片上洛朗系数的关联完全公式：极点阶数等于具有非零总系数的最大阶数，而非单个旗标的最大阶数。一般公式恢复了经典的$A_2$和$A_3$奇异数据、赵的欧拉 - 扎吉尔留数公式和秋山 - 江见 - 谷川列表以及$C_2$和$G_2$留数函数。对于$B_3$和$C_3$，我们推导正对角留数的载体几何和低阶分解，确定在$s = 1/8$处的三项抵消，并表明负半整数是唯一可能的双极点位置，恢复了已知在$s = -1/2$处的$B_3$双系数。
英文摘要
Let $Φ$ be an irreducible crystallographic root system, and let $Z_Φ(\mathbf{s})$ denote the untwisted Komori-Matsumoto-Tsumura zeta function with one exponent for each positive coroot. For a nonempty set $S$ of simple nodes, let $H_{S,\ell}$ be the hyperplane on which the exponents of the roots meeting $S$ sum to $|S|-\ell$. We prove that every proper-support hyperplane $H_{S,\ell}$ is a genuine polar divisor at a generic point, whereas exact homogeneity leaves only the unshifted full-support divisor. The residue on $H_{S,\ell}$ is expressed as a finite Taylor-jet sum of reduced projective periods and polynomially weighted complementary root-system zeta functions. On the maximal support wonderful model, boundary terms are indexed by strict decorated flags. We derive recursive flag residues, component-mass gamma factors, and an incidence-complete formula for the Laurent coefficients on any transverse affine slice. In particular, the pole order is determined by the first nonzero aggregate coefficient, not by the largest order of an individual flag. The general formulas recover the classical $A_2$ and $A_3$ singular data, Zhao's Euler-Zagier residues, and the rank-two $C_2$ and $G_2$ residue functions. For $B_3$ and $C_3$ we derive the carrier geometry and the lower-rank factorizations of the positive residues, identify the three-term cancellation at $s=1/8$, and show that negative half-integers are the only possible locations of double poles. The known $B_3$ double coefficient at $-1/2$ is recovered in the flag normalization.
