Let \(G\) be a finite group, \(k\) an arbitrary field and \(k(t)\) the rational function field over \(k\) with \(n\) indeterminates \( t = (t_{1}, t_{2},..., t_{n})\). A polynomial \(f_{t}^{G}(X) \in k(t)[X]\) is called k-generic for \(G\) if the Galois group of \(f_{t}^{G}(X)\) over \(k(t)\) is isomorphic to \(G\) and every \(G\)-Galois extension \(L/M\), where \(k \subset M\) and \(M\) is infinite, can be obtained as \(L = \text{Spl}_{M} f_{a}(X)\) the splitting field of \(f_{a}(X)\) over \(M\) for some \( a = (a_{1}, a_{2},..., a_{n}) \in M^{n}\).
The authors are interested to determine, for a field \(M\) containing \(k\) and \(a, b \in M^{n}\), the intersection of \(\text{Spl}_{M} f_{a}^{G}(X)\) and \(\text{Spl}_{M} f_{b}^{G}(X)\). In this paper, the authors study a method to give an answer to the intersection problem of \(k\)-generic polynomials via Tschirnhausen transformation and multi-resolvent polynomials. In particular, the authors study the case of quintic generic polynomials with two parameters and where \(G\) is a cyclic or dihedral group of degree \(5\), or the Frobenius group of order \(20\).