Degree bounds for fields of rational invariants of \(\mathbb{Z}/p\mathbb{Z}\) and other finite groups (Q6544506)

Let \(G\) be a finite group, \(\mathbf{k}\) a field of characteristic prime to \(|G|\), and \(V\) a finite dimensional representation of \(G\) over \(\mathbf{k}\). The goal of this paper is to give degree bounds for field generators of the field of rational invariants \(\mathbf{k}(V)^G\). This is closely related to a popular topic of invariant theory, namely, the study of degree bounds for algebra generators of invariant rings. The inquiry deals with abelian groups focusing on the case of the cyclic group \(\mathbb{Z}/p\mathbb{Z}\) of prime order.\N\NAnalogously to the definition of the Noether number \(\beta(G,V)\) of the representation \(V\), one can define the number \(\beta_{\mathrm{field}}(G,V)\) as the minimum degree of polynomial invariants needed to generate the field of invariant rational functions \(\mathbf{k}(V)^G\) as a field extension of \(\mathbf{k}\). A similar quantity is \(\gamma_{\mathrm{field}}(G,V)\), the minimal number \(d\), such that \(\mathbf{k}(V)^G\) has a transcendence basis of polynomials of degree \(\le d\). The aim of the paper is to give lower bounds for \(\gamma_{\mathrm{field}}(G,V)\) and upper bounds for \(\beta_{\mathrm{field}}(G,V)\) (of course, \(\gamma_{\mathrm{field}}(G,V)\le \beta_{\mathrm{field}}(G,V)\) always holds).\N\NThe main results are as follows. Suppose that \(G\) is a finite abelian group and \(V\) a finite dimensional, non-modular faithful representation containing exactly \(m\) distinct nontrivial characters of \(G\). Then \(\gamma_{\mathrm{field}}(G,V)\geq \sqrt[m]{|G|}\) (Theorem 3.1). (A similar bound for \(\gamma_{\mathrm{field}}(G,V)\) is obtained for any faithful finite dimensional representation \(V\) of an arbitrary finite group \(G\) in Theorem 3.2). The most important part of the inquiry focuses on the case of the cyclic group \(\mathbb{Z}/p\mathbb{Z}\) of prime order. Theorem 3.11 asserts that if \(V\) a finite dimensional, non-modular representation of \(\mathbb{Z}/p\mathbb{Z}\) containing at least \(3\) distinct nontrivial characters, then \(\beta_{\mathrm{field}}(\mathbb{Z}/p\mathbb{Z},V)\leq\frac{p+3}{2}\). However, this is not a sharp bound. The sharp upper bound is believed to be \(\left\lceil\frac{p}{\lceil m/2 \rceil}\right\rceil\), where \(m\) is the number of distinct, nontrivial characters occurring in the representation \(V\) (Conjecture 5.1).