Modular Techniques in Computational Algebraic Geometry

Computations over the rational numbers often suffer from intermediate coefficient growth. One approach to this problem is to determine the result modulo a number of primes and then lift to the rationals. This method is guaranteed to work if we use a sufficiently large set of good primes. In many applications, however, there is no efficient way of excluding bad primes. We develop a new technique for rational reconstruction which will nevertheless return the correct result, provided the number of good primes in the set is large enough. We discuss applications of this technique in computational algebraic geometry. This is joint work with Claus Fieker, Wolfram Decker, Gerhard Pfister, and Santiago Laplagne.