Help Privacy Policy Disclaimer
  Advanced SearchBrowse




Journal Article

Riemann Hypothesis for DAHA superpolynomials and plane curve singularities


Cherednik,  Ivan
Max Planck Institute for Mathematics, Max Planck Society;

External Resource
Fulltext (restricted access)
There are currently no full texts shared for your IP range.
Fulltext (public)

(Preprint), 696KB

Supplementary Material (public)
There is no public supplementary material available

Cherednik, I. (2018). Riemann Hypothesis for DAHA superpolynomials and plane curve singularities. Communications in Number Theory and Physics, 12(3), 409-490. doi:10.4310/CNTP.2018.v12.n3.a1.

Cite as: https://hdl.handle.net/21.11116/0000-0003-AA7D-2
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-St\"ohr zeta-functions are discussed.