Riemann Hypothesis for DAHA superpolynomials and plane curve singularities

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 bar right arrow qt satisfy the Riemann Hypothesis for sufficiently small q for uncolored algebraic knots, presumably for q = 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-Stohr zeta-functions are discussed.