Invariants of finite groups over fields of characteristic \(p\) (Q1354655)

Let \(V\) a finite-dimensional \(K\) vector space and let \(G\) be a finite group of \(K\)-linear automorphisms of \(V\). Consider \(G\) acting on the symmetric algebra \(K [V^m]\) of \(V^m\) by the diagonal action on \(V^m\). A result of Noether implies that, if \(\text{char} K =0\), then \(K[V^m]^G\) can be generated as a \(K\)-algebra by polynomials whose degrees are \(\leq|G|\), no matter how large \(m\) is. The paper under review proves that this result no longer holds when the characteristic of \(K\) divides \(|G|\). More precisely, it is proved in this case that there is a positive number \(\alpha\), depending only on \(|G|\) and \(\text{char} K\), such that every set of \(K\)-algebra generators of \(K[V^m]^G\) contains a generator whose degree is \(\geq \alpha m\).