Polynomial bounds for rings of invariants (Q2701610)

This article gives a polynomial degree bound for invariant rings of linearly reductive groups. More precisely, let \(G\) be a linearly reductive group acting rationally and linearly on a vector space \(V\) of finite dimension~\(n\). By a theorem of Hilbert, it is known that the invariant ring \({\mathcal O}(V)^G\) is a finitely generated algebra over the ground field \(K\). It is an interesting question to determine the minimal number \(\beta\left({\mathcal O}(V)^G\right)\) such that \({\mathcal O}(V)^G\) is generated by homogeneous invariants of degree at most \(\beta\left({\mathcal O}(V)^G\right)\). \textit{V. L. Popov} gave the following upper bound: NEWLINE\[NEWLINE \beta\left({\mathcal O}(V)^G\right) \leq n \cdot \text{LCM}\{1,2, \ldots ,\sigma(G,V)\} NEWLINE\]NEWLINE [Izv. Akad. Nauk SSSR, Ser. Mat. 45, 1100-1120 (1981; Zbl 0478.14006)]. Here \(\sigma(G,V)\) is the minimal integer~\(d\) such that there exist homogeneous invariants of degree at most~\(d\) whose variety is the nullcone. The least common multiple in Popov's bound comes from the fact that he took homogeneous invariants giving the nullcone and substituted them by powers to make the degrees equal. This is necessary to apply Noether normalization in order to obtain a homogeneous system of parameters. NEWLINENEWLINENEWLINEDerksen's main new idea is that any set of homogeneous invariants giving the nullcone can be made into a homogeneous system of parameters of a bigger ring, preserving the original degrees. This allows him to get away without forming the LCM, and leads to the bound NEWLINE\[NEWLINE \beta\left({\mathcal O}(V)^G\right) \leq \max\left\{2,\tfrac{3}{8} n \cdot \sigma(G,V)^2\right\}. NEWLINE\]NEWLINE Although this bound is usually not sharp, it constitutes a great advance. In the last section of the paper an effective bound for \(\sigma(G,V)\) is given, which is polynomial in the degrees of the polynomials defining \(G\) and its action on \(V\).