Dimension polynomials of extensions of difference fields
Let \(F\) be a difference field with the basic set of translations \(\Delta =\{\alpha_1,\ldots,\alpha_m\}\), \(\text{char}\, F=0\), \(\Gamma =\{\gamma =\alpha_1^{k_1}\cdots\alpha_m^{k_m} \mid k_1,\ldots,k_m\in\mathbb Z\}\) and \(\Gamma(s)=\{\gamma \in \Gamma \mid \text{ord}\,\gamma =\sum^{m}_{i=1}| k_i| \leq s\}\). The main result is the following: for any elements \(\eta_1,\ldots,\eta_n\) of an inversive difference field extension of \(F\) there exists an integral polynomial \(\psi\) such that for any sufficiently large integer \(s\) the transcendence degree of \(F((\gamma \eta_j),\;\gamma \in \Gamma(s),\;1\leq j\leq n)\) over \(F\) is equal to \(\psi(s)\). The polynomial \(\psi\) is called the difference dimension polynomial of the extension \(F(\eta)\) over \(F\). Some properties of dimension polynomials are described.
scientific article; zbMATH DE number 3863323