Generic Polynomials with Few Parameters

We call a polynomial g(t1, . . . , tm, 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 L and Gal(N/L) G arises as the splitting field of a suitable specialization g(1, . . . , m, X) with 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 formulas for generic polynomials for the groups GU2(q2 ) and GO3(q).

• Type: Journal
• Year: 2000
• Where: JSC
• Authors: Gregor Kemper, Elena Mattig
• Keywords: Faithful Linear Representation, Generic Polynomials, JSC 2000, Polynomials