A fundamental problem in numerical computation and computational geometry is to determine the sign of arithmetic expressions in radicals. Here we consider the simpler problem of deciding whether is zero for given rational numbers A i , C i , X i . It has been known for almost twenty years that this can be decided in polynomial time [2] . In this paper we improve this result by showing membership in uniform TC 0 . This requires several significant departures from Blömer's polynomial-time

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... m as the latter crucially relies on primitives, such as gcd computation and binary search, that are not known to be in TC 0 .