Groups and Algebraic Complexity (Abstract)

In recent years, concepts from group theory have played an important role in the derivation of lower bounds for computational complexity. In this talk we present two new results in algebraic complexity obtained with group-theoretical arguments. We show that any algebraic computation tree for the membership question of a compact set S in Rn must have height Ω(log(βi(S)))−cn for all i, where βi are the Betti numbers. We also show that, to compute the sum of n independent radicals using logarithms, exponentiations and root-takings, at least n operations are required. (The second result was obtained jointly with Dima Grigoriev and Mike Singer.) This talk will be self-contained.