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Noether’s bound for polynomial invariants of finite groups. (English) Zbl 0967.13004

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 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 \(G\).

MSC:

13A50 Actions of groups on commutative rings; invariant theory
20C15 Ordinary representations and characters
20K15 Torsion-free groups, finite rank

Citations:

Zbl 0770.20004
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