Generic Polynomials with Few Parameters

We call a polynomial g(t₁, . . . , t_m,X ) over a field K generic for a group G if it has Galois group G as a polynomial inX , and if every Galois field extension N / L withK L and Gal(N / L) ≤ G arises as the splitting field of a suitable specializationg (λ₁, . . . , λ_m, X) withλ_i 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 GU₂(q²) andGO₃ (q).