On the complexity of algebraic numbers, and the bit-complexity of straight-line programs¹

We investigate the complexity of languages that correspond to algebraic real numbers, and we present improved upper bounds on the complexity of these languages. Our key technical contribution is the presentation of improved uniform TC0 circuits for division, matrix powering, and related problems, where the improvement is in terms of "majority depth" (initially studied by Maciel and Therien). As a corollary, we obtain improved bounds on the complexity of certain problems involving arithmetic circuits, which are known to lie in the counting hierarchy, and we answer a question posed by Yap.