For any polynomial ideal I with given generators, testing whether another
given polynomial p belongs to I or the radical of I is equivalent to
determining whether p is equal to 0 modulo I or the radical of I. The
test of ideal and radical ideal membership is a fundamental problem
in computational polynomial ideal theory and algebraic geometry with
diverse applications. In this talk, we show how to solve the problem
algorithmically and effectively by using the methods of Groebner bases
and triangular decomposition. We discuss in particular the application
of membership test to automated geometric theorem proving.