On Computing Algebraic Functions Using Logarithms and Exponentials

Dima Grigoriev, Michael F. Singer, Andrew Chi-Chih Yao

Abstract: Let $\rho$ be a set of algebraic expressions constructed with radicals and arithmetic operations, and which generate the splitting field F of some polynomial. Let $N_{\beta}(\rho)$ be the minimum total number of root-takings and exponentiations used in any straightline program for computing the functions in $\rho$ by taking roots, exponentials, logarithms, and performing arithmetic operations. In this paper it is proved that $N_{\beta}(\rho) = v(G)$, where $v(G)$ is the minimum length of any cyclic Jordan-Hölder tower for the Galois group G of F. This generalizes a result of Ja’Ja’ [Proceedings of the 22nd IEEE Symposium on Foundations of Computer Science, 1981, pp. 95–100], and shows that the inclusion of certain new primitives, such as taking exponentials and logarithms, does not improve the cost of computing such expressions as compared with programs that use only root-takings.