THE COMPUTATIONAL COMPLEXITY OF ALGEBRAIC NUMBERS

Let ix^j be a sequence approximating an algebraic number a of degree r, and let ^ +^ m ctKx^x^ ,... »x^_^j) , for some rational function co with integral coefficients. Let M denote the number of multiplications or divisions needed to compute cp and let M denote the number of multiplications or divisions, except by constants, needed to compute cp. Define the multiplication log2p efficiency measure of {x^} as E({x}) — — or as log2p L E({x^}) « — g — , where p is the order of convergence of (x i). Rung [1] showed that E^x^}) £ 1 or equivalently, M £ log2p. In this paper we show that (i) M * log2[r(fpl-D + 1] 1; (ii) if (C x i}) s 1 then a is a rational number; (iii) if E({xi}) = 1 then a is a rational or quadratic irrational number. This settles the question of when the multiplication efficiency E^x^}) or ECfx^)) achieves its optimal value of unity.