Polynomial-time presentations of algebraic number fields

Abstract

© 2018, Springer International Publishing AG, part of Springer Nature. Using an extension of the notion of polynomial time presentable structure we show that some natural presentations of the ordered field ℝalgof algebraic reals and of the field ℂalgof 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 ℂalgand for the problem of root-finding for polynomials in ℂalg[x] which improve the previously known bound.