An invariant for difference field extensions (Q439213)

The paper under review introduces a new invariant of an ordinary difference field extension, called the \textit{distant degree} of the extension, and studies properties of this invariant. Let \(K\) be an ordinary difference field with a distinguished endomorphism \(\sigma\). If \(a\) is a finite tuple in some difference field extension of \(K\) such that \(\sigma(a)\) lies in the algebraic closure of \(K\), then the distant degree and inverse distant degree of \(a\) over \(K\) are defined as \(dd(a/K) = \lim_{k\rightarrow\infty}[K(a, \sigma^{k}(a)):K(a)]^{1/k}\) and \(idd(a/K) = \lim_{k\rightarrow\infty}[K^{\text{inv}}(a, \sigma^{-k}(a)):K^{\text{inv}}(a)]^{1/k}\), respectively. It is shown that if \(a\) and \(b\) are tuples over \(K\) (the coordinates of \(a\) and \(b\) lie in some large difference overfield \(\mathcal{U}\) of \(K\)) such that the algebraic closure \(K(a)_{\sigma}^{\text{alg}}\) of the difference field \(K(a)_{\sigma}\) generated by \(a\) over \(K\) coincides with \(K(b)_{\sigma}^{alg}\), then \(dd(a/K) = dd(b/K)\). Also, for any tuple \(a\) over \(K\), \(dd(a/K) = dd(a/K^{inv})\) and \(dd(a/K)ild(a/K) = idd(a/K)ld(a/K)\) where \(ld(a/K)\) and \(ild(a/K)\) denote the limit degree and inverse limit degree of \(K(a)_{\sigma}\) over \(K\). (The limit degree and inverse limit degree are characteristics of a difference field extension introduced in [\textit{R. M. Cohn}, ``An invariant of difference field extensions'', Proc. Am. Math. Soc. 7, 656--661 (1956; Zbl 0070.27001)], see also Chapter 5 of [\textit{R. M. Cohn}, Difference algebra. New York etc.: Interscience Publishers (1965; Zbl 0127.26402)] and Chapter 4 of [\textit{A. Levin}, Difference algebra. Algebra and Applications 8. New York, NY: Springer (2008; Zbl 1209.12003)]). Furthermore, the authors prove that there exists a constant \(m = m(a, K)\) such that \(dd(a/K) = \frac{ld(a/K)}{m}\) (assuming \(\sigma(a)\in K(a)^{\text{alg}}\) and \(K(a, \sigma(a)):K(a) = ld(a/K)\)), and establish some other properties of the distant degree and its relation to the limit degree. In particular, they prove that if \(a\) and \(b\) are two tuples in \(\mathcal{U}\supseteq K\) such that \(\sigma(a)\in K(a)^{alg}\), \(\sigma(b)\in K(b)^{\text{alg}}\), then \(dd(a,b/K)\geq dd(a/K(b)_{\sigma})dd(b/K)\) and if \(b\in K(a)^{\text{alg}}\), then \(dd(b/K)\geq dd(a/K)\). It is also shown that the first of these inequalities can be strict, so the distant degree is not multiplicative in towers of extensions. The proofs are largely based on the following result (Theorem 1.8): if \(K = \sigma(K)\) and \(a\) is a tuple such that \(\sigma(a)\in K(a)^{alg}\), then there exists \(c\in K(a)_{\sigma}\) such that \(a\in K(c)^{alg}\) and \(\sigma^{i}(c)\in K(c, \sigma^{i}(c))\) for every \(l>i>0\).NEWLINENEWLINEThe last part of the paper considers the obtained information on distant degree in the context of difference subgroups of algebraic groups. In this part the authors establish a relationship between their results on difference fields (in the case of zero characteristic) and the results on totally disconnected locally compact groups obtained in [\textit{G. Willis}, ``The structure of totally disconnected, locally compact groups'', Math. Ann. 300, No. 2, 341--363 (1994; Zbl 0811.22004) and ``Further properties of the scale function on a totally disconnected group'', J. Algebra 237, No. 1, 142--164 (2001; Zbl 0982.22001)]. This relationship allows the authors to prove some new interesting statements on difference field extensions. In particular, they show that if a tuple \(a\) over \(K\) satisfies the condition \(K(a, \sigma(a)):K(a) = ld(a/K)\) and \(\tau\in \Aut\left(K(a)^{\text{alg}}/K(a)\right)\), then the difference fields \((K(a)_{\sigma}, \sigma)\) and \((K(a)_{\sigma\tau}, \sigma\tau)\) are isomorphic by a \(K\)-isomorphism taking \(a\) to \(a\).