Asymptotic Behavior of Kac Polynomials

Abstract

We conjecture a formula supported by computations for the valuation of Kac polynomials of a quiver, which only depends on the number of loops at each vertex. We prove a convergence property of renormalized Kac polynomials of quivers when increasing the number of arrows: they converge in the ring of power series, with a linear rate of convergence. Then, we propose a conjecture concerning the global behavior of the coefficients of Kac polynomials. All computations were made using SageMath.