Noether's bound for polynomial invariants of finite groups
If \(G\) is a finite group acting linearly on a finite dimensional complex vector space \(V\), then there is an induced action of \(G\) on the polynomial algebra \(\mathbb{C}[V]\) which is the direct sum of the symmetric tensor powers of \(V\). The authors prove: If \(G\) is the semi-direct product of cyclic groups of odd prime order, then the algebra \(C[V]^G\) of polynomial invariants is generated by its elements whose degree is bounded by \({5\over 8}|G|\). By this the authors sharpen the improved bound for Noether's theorem due to \textit{B. Y. Schmid} in: Topics in invariant theory, Proc. Sémin. Algèbre, Paris 1989-1990, Lect. Notes Math. 1478, 35-66 (1991; Zbl 0770.20004). They show: \(C[V]^G\) is generated by elements of degree \(\leq{3\over 4}|G|\) for any non-cyclic group \(G\).
action of semi-direct product of cyclic groups
scientific article; zbMATH DE number 1449195