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The use of Bad Primes in Rational Reconstruction

A standard method for finding a rational number from its values modulo a collection of primes is to determine its value modulo the product of the primes via Chinese remaindering, and then use Farey sequences for rational reconstruction. Successively enlarging the set of primes if needed, this method is guaranteed to work if we restrict ourselves to good primes. Depending on the particular application, however, there may be no efficient way of identifying good primes. In the algebraic and geometric applications we have in mind, the final result consists of an a priori unknown ideal (or module) which is found via a construction yielding the (reduced) Groebner basis of the ideal. In this context, we discuss a general setup for modular and, thus, potentially parallel algorithms which can handle bad primes. A key new ingredient is an error tolerant algorithm for rational reconstruction via Gaussian reduction.