DIMACS TR: 93-07

On Computing Algebraic Functions Using Logarithms and Exponentials

Authors: Dina Grigoriev, Michael Singer, and Andrew Yao


Let f be a set of algebraic expressions constructed with radicals and arithmetic operations, and which generate the splitting field F of some polynomial. Let N_b(f) be the minimum total number of root-takings and exponentiations used in any straightline program for computing the functions in f by taking roots, exponentials, logarithms, and performing arithmetic operations. In this paper it is proved that N_b(f) = g(G), where g(G) is the minimum length of any cyclic Jordan-Hoelder tower for the Galois group G of F. This generalizes a result of Ja'Ja' [1], 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 which use only root-takings.

Paper available at: ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1993/93-07.ps
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