Generic polynomials with few parameters

We call a polynomial g(t(1),...,t(m),X) over a field K generic for a group G if it has Galois group G as a polynomial in X, and if every Galois field extension N/L with K subset of or equal to L and Gal(N/L) less than or equal to G arises as the splitting field of a suitable specialization g(lambda (1),...,lambda (m),X) with lambda (i) is an element of L. We discuss how the rationality of the invariant field of a faithful linear representation leads to a generic polynomial which is often particularly simple and therefore useful. Then we consider various examples and applications in characteristic 0 and in positive characteristic. These include results on so-called vectorial polynomials and a generalization of an embedding criterion given by Abhyankar. We give recursive formulas for generic polynomials over a field of defining characteristic for the groups of upper unipotent and upper triangular matrices, and explicit formulae for generic polynomials for the groups GU(2)(q(2)) and GO(3)(q). (C) 2000 Academic Press.