Mahler and Koksma classification of points in \(\mathbb R^n\) and \(\mathbb C^n\)
In order to classify the transcendental numbers, Mahler and Koksma introduced exponents of approximation, \(\omega_d(\xi)\) and \(\omega_d^*(\xi)\), for each real or complex number \(\xi\). The paper under review deals with higher dimensional variants of these quantities and classifications. Let \({\mathcal P}_d\) denote the set of non-zero integer polynomials in \(n\) variables with total degree at most \(d\). Also, let \(|P|\) denote the maximum modulus among the coefficients of \(P\). For a point \(\xi \in {\mathbb C}^n\), \(\psi_d(\xi)\) is defined to be the supremum of all \(\psi\) for which \[ 0 < | P(\xi) | / | P | < | P |^{-\psi} \] has infinitely many solutions \(P \in {\mathcal P}_d\). Similarly, \(\psi_d^*(\xi)\) is defined to be the supremum of all \(\psi^*\) for which there are infinitely many \(P \in {\mathcal P}_d\) with a zero \(\alpha \in {\mathbb C}^n\) such that \(P(\xi) \neq 0\) and \[ | \xi - \alpha | < | P |^{-\psi^*}. \] Here, the absolute value on the left hand side is the maximum modulus of the coordinates of \(\xi - \alpha\). When \(n = 1\), \(\psi_d(\xi) = \omega_d(\xi) + 1\) and \(\psi_d^*(\xi) = \omega_d^*(\xi) + 1\), where \(\omega_d(\xi)\) and \(\omega_d^*(\xi)\) are the exponents defined by Mahler and Koksma respectively. In the present paper, a classification of points in \({\mathbb C}^n\) and \({\mathbb R}^n\) is given by the new exponents in analogy with the classifications of Mahler and Koksma. We call two points \(\eta \in {\mathbb C}^n\) or \({\mathbb R}^n\) and \(\xi \in {\mathbb C}^l\) or \({\mathbb R}^l\) algebraically related if \(\overline{{\mathbb Q}(\eta)} =\overline{{\mathbb Q}(\xi)}\), \textit{i.e}, if the algebraic closures of \({\mathbb Q}(\eta)\) and \({\mathbb Q}(\xi)\) coincide. It was shown by \textit{K. Yu} [J. Reine Angew. Math. 377, 113--126 (1987; Zbl 0614.10031)] that points which are algebraically related are in the same class in the generalized Mahler sense, \textit{i.e.}, in the classification given by the \(\psi_d\) exponents. This result is reproved in the present paper. Additionally, it is shown that if \(\xi, \eta \in {\mathbb C}^n\) have algebraically independent components and are algebraically related, then they are in the same class in the generalized Koksma sense, \textit{i.e.}, in the classification given by the \(\psi_d^*\) exponents. Additional metrical results on the exponents are also proved, and a number of open problems and conjectures are given, suggesting that the higher dimensional setting is much more intricate than the one-dimensional case. For instance, it is conjectured that there are T-points, which are also S\(^*\)-points whenever \(n > 1\). This is not the case when \(n = 1\).
scientific article; zbMATH DE number 5135178