Modular algorithms based on the Chinese Remainder Theorem (CRT) control intermediate expression growth by performing computations modulo small primes. Some primes, that we call "unlucky," produce incorrect results, or no result; their number can be bounded by finding a nonzero \(U \in \mathbb {Z}\) with the property that all unlucky primes divide U.

We present a full analysis of the bit complexity of an efficient algorithm for the computation of at least one point in each connected component of a smooth real hypersurface. This is a basic and important operation in semi-algebraic geometry: it gives an upper bound on the number of connected components of a real hypersurface, and is also used in many higher level algorithms.