On subgroups of the additive group in differentially closed fields
The author studies additive subgroups of a differentially closed field of characteristic zero with \(m\) commuting derivations (\(m\)-DCF\(_0\)), i.e., subgroups of some Cartesian power of the additive group of the field. This was first studied by \textit{T. McGrail} [J. Symb.\ Log.\ 65, No. 2, 885--913 (2000; Zbl 0960.03031)], and later by \textit{R. Moosa, A. Pillay} and \textit{T. Scanlon} [J. Reine Angew.\ Math.\ 620, 35--54 (2008; Zbl 1223.12007)], who showed that any non-modular regular type must be non-orthogonal to the generic type of an additive subgroup.NEWLINENEWLINEThe author first quickly reviews the properties of the dimension polynomial: If \(K\) is a model of \(m\)-DCF\(_0\) and \(\eta\) a tuple in some elementary extension, then there is a polynomial NEWLINE\[NEWLINE\omega_{\eta/K}(X)=\sum_{i=0}^\tau a_\tau{W+i\choose i}NEWLINE\]NEWLINE with \(a_\tau\not=0\) and \(\tau\leq m\), such that \(\omega_{\eta/K}(s)\) is the transcendence degree \([K(\delta^{\bar\alpha}\eta:|\bar\alpha|\leq s):K]\) for sufficiently large \(s\). Then \(a_m\) (which may be zero) is the \(\Delta\)-transcendence degree of \(K\langle\eta\rangle\) over \(K\). We call \(\tau\) the \(\Delta\)-type of \(\eta\) over \(K\) and \(a_\tau\) the typical \(\Delta\)-dimension. If \(X\) is an irreducible \(\Delta\)-variety, its \(\Delta\)-type and typical \(\Delta\)-dimension are that of a generic point. A \(\Delta\)-group is \textit{\(\Delta\)-type minimal} if it has no proper subgroup of the same \(\Delta\)-type.NEWLINENEWLINEThe author shows that any two \(\Delta\)-type minimal additive subgroups with non-orthogonal generic types have isomorphic proper quotients, and in particular the same \(\Delta\)-type and typical \(\Delta\)-dimension. An interpretable field is definably isomorphic to a definable subfield; it is \(\Delta\)-type minimal of typical \(\Delta\)-dimension \(1\). Any two non-isomorphic interpretable subfields are orthogonal. A \(\Delta\)-type minimal subgroup of the additive group is non-orthogonal to a field if and only if its typical \(\Delta\)-dimension is \(1\).NEWLINENEWLINENext, for every \(k<\omega\) an example is given of a type \(p_k\) of Lascar rank \(\omega\), \(\Delta\)-type \(1\) and typical \(\Delta\)-dimension \(k\). This contradicts in particular a rank inequality claimed by McGrail.NEWLINENEWLINEFinally, the author shows that the generic type of a quotient of the heat variety must be locally modular, and answers negatively a question of Cassidy on the existence of a certain Jordan-Hölder series for the heat variety.
scientific article; zbMATH DE number 6047767