Polynomial-Time Presentations of Algebraic Number Fields

Using an extension of the notion of polynomial time presentable structure we show that some natural presentations of the ordered field \({\mathbb R}_{\text{ alg }}\) of algebraic reals and of the field \({\mathbb C}_{\text{ alg }}\) of algebraic complex numbers are polynomial-time equivalent to each other and are polynomial time. We also establish upper complexity bounds for the problem of rational polynomial evaluation in \({\mathbb C}_{\text{ alg }}\) and for the problem of root-finding for polynomials in \({\mathbb C}_{\text{ alg }}[x]\) which improve the previously known bound.