Diophantine approximation and nonstandard ultrametric distances
In his paper ``Classification de Mahler et distances locales'' [Bull. Aust. Math. Soc. 49, No. 2, 219--238 (1994; Zbl 0799.11019)], \textit{P. Philippon} considered the quotient space \(KM_1\) of \(\mathbb C\) by the equivalence relation \(\alpha\sim\beta\) if \(\alpha\) is algebraic over \({\mathbb Q}(\beta)\) and \(\beta\) is algebraic over \({\mathbb Q}(\alpha)\). This relation classifies complex numbers according to algebraic independence. Here, the author works out a similar theory for two other quotient spaces. The first one \(QH_1\) classifies real numbers according to rational dependence, the equivalence relation is \(\alpha\sim\beta\) if \(\beta=(a\alpha+b)/(c\alpha+d)\) with rational integers \(a,b,c,d\) satisfying \(ad-bc\not=0\). The second one \(QL_1\) classifies real numbers according to linear dependence, the equivalence relation is \(\alpha\sim\beta\) if \(q\beta=p\alpha\) with non-zero rational integers \(p,q\). She considers not only the classical case of real numbers, but also the field of formal Laurent series in one variable in any characteristic. She defines uniform topologies on these quotient sets which reflect Diophantine properties.
scientific article; zbMATH DE number 5016732
scientific article