The constants of Lotka-Volterra derivations
Let \(R = K[x_{1},\dots, x_{n}]\) be a polynomial ring over a field \(K\) with characteristic zero. Given parameters \(C_{i}\in K\) (\(1\leq i\leq n\)), the Lotka-Volterra (\(K\)-linear) derivation \(d\) of \(R\) is defined on the generators as follows: \(d(x_{i}) = x_{i}(x_{i-1}-C_{i}x_{i+1})\) where the indexing is circular, that is, \(n+e\) and \(e\) are identified for every \(e\in \mathbb{Z}\). Let \(f = \sum_{i=1}^{n}(\prod_{j=1}^{i-1}C_{j})x_{i}\in R\) and let a set \(\emptyset\neq \mathcal{A}\subseteq \mathbb{Z}_{n}\) be closed under the mapping \(i\mapsto i+2\). If \(n\) is odd, then \(\mathcal{A} = \mathbb{Z}_{n}\); for \(n\) even, there are two additional proper subsets: \(\mathcal{E} = \{2i\,: i\leq n/2\}\) and \(\mathcal{O} = \{2i-1\,: i\leq n/2\}\). If \(C_{i}\), \(i\in\mathcal{A}\), are positive rational numbers such that \(\prod_{i\in\mathcal{A}}C_{i} =1\), then there exist unique coprime integers \(\theta_{i}\), \(i\in\mathcal{A}\), such that \(\theta_{i+2} = C_{i}\theta_{i}\). Let \(g_{\mathcal{A}} =\prod_{i\in\mathcal{A}}x_{i}^{\theta_{i}}\) and \(\mathcal{A}' =\{i+1\,:\,i\in\mathcal{A}\}\). One of the two main results of the paper under review is the following theorem. Theorem 1.1. With the above notation, the ring of constants \(R^{d}=\{f\in R\,: d(f) = 0\}\) is a polynomial algebra. If not all \(C_{i}\) are equal to \(1\) and \(n>4\), then the number of generators of \(R^{d}\) is equal to (i) \(0\) if \(\prod C_{i}\neq 1\) and no \(g_{\mathcal{A}}\) is defined; (ii) \(3\) if \(n\) is even and both \(g_{\mathcal{E}}\) and \(g_{\mathcal{O}}\) are defined; (iii) \(2\) if \(n\) is odd and \(g_{\mathbb{Z}_{n}}\) is defined, or \(n\) is even and \(\prod C_{i} =1\) but only one of \(g_{\mathcal{E}}\) and \(g_{\mathcal{O}}\) is defined; (iv) \(1\) if \(\prod C_{i} =1\) but no \(g_{\mathcal{A}}\) is defined, or \(n\) is even and \(\prod C_{i}\neq 1\) but only one of \(g_{\mathcal{E}}\) and \(g_{\mathcal{O}}\) is defined. The other main theorem proved in the paper is a version of Theorem 1.1 for the case \(n=4\).
scientific article; zbMATH DE number 6599111