It is known that given a singular Hankel matrix \(H\), the kernel of \(H\) consists of all polynomials of the form \(m(x) p(x)\), where \(m\) is an arbitrary polynomial whose degree does not exceed a certain bound. The polynomial \(p\), which is the greatest common divisor of all polynomials in the kernel of \(H\), and the bound \(b\) for the degree of \(m\) are the characteristics of \(H\) referred to in the title. The author presents an approach to characteristic numbers of Hankel matrices based on applications of the infinite companion matrix. His results are consequences of a strengthening of a classical result of Frobenius. For the definitions, see \textit{G. Heinig} and \textit{K. Rost} [Algebraic methods for Toeplitz-like matrices and operators (1984; Zbl 0549.15013)].