关于DAHA超多项式和平面曲线奇点的黎曼假设
Riemann Hypothesis for DAHA superpolynomials and plane curve singularities
摘要：
代数纽结的稳定Khovanov-Rozansky多项式期望与嵌套Hilbert格式的某些生成函数、超多项式和相应平面曲线奇点的标记Jacobian因子相一致。此外，这3个族在理论上匹配DAHA超多项式。这些超多项式可以被认为是Hasse-Weil zeta函数的奇异对应和推广。我们猜想，所有的$a$ -系数的DAHA超多项式的替代$q\mapsto qt$满足黎曼假设充分小的$q$无色代数结，大概为$q\le 1/2$作为$a=0$。这至少对于$a=0$可以部分地扩展到代数链路。彩色链接也被考虑，虽然主要是矩形杨图。与Kapranov的动机zeta和Galkin Stöhr zeta函数的连接进行了讨论。
英文摘要：
Stable Khovanov-Rozansky polynomials of algebraic knots are expected to coincide with certain generating functions, superpolynomials, of nested Hilbert schemes and flagged Jacobian factors of the corresponding plane curve singularities. Also, these 3 families conjecturally match the DAHA superpolynomials. These superpolynomials can be considered as singular counterparts and generalizations of the Hasse-Weil zeta-functions. We conjecture that all $a$-coefficients of the DAHA superpolynomials upon the substitution $q\mapsto qt$ satisfy the Riemann Hypothesis for sufficiently small $q$ for uncolored algebraic knots, presumably for $q\le 1/2$ as $a=0$. This can be partially extended to algebraic links at least for $a=0$. Colored links are also considered, though mostly for rectangle Young diagrams. Connections with Kapranov's motivic zeta and the Galkin-Stöhr zeta-functions are discussed.
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