arXiv subjects
Publications and source records attributed to Christian d'Elbée.
It is shown that every dp-minimal integral domain $R$ is a local ring and for every non-maximal prime ideal $\mathfrak p $ of $R$, the localization $R_{\mathfrak p }$ is a valuation ring and $\mathfrak{p}R_{\mathfrak{p}}=\mathfrak{p}$. Furthermore, a dp-minimal integral domain is a valuation ring if and only if its residue field is infinite or its residue field is finite and its maximal ideal is principal.
Let $A$ be an abelian variety in a field of characteristic $0$. We prove that the expansion of $A$ by a generic divisible subgroup of $A$ with the same torsion exists provided $A$ has few algebraic endomorphisms, namely $\mathrm{End}(A)=\mathbb Z$. The resulting theory is $\mathrm{NSOP}_1$ and not simple. Note that there exist abelian varieties $A$ with $\mathrm{End}(A) = \mathbb{Z}$ of any genus. We indicate how this result can be extended to any simple abelian variety by considering the expansion by a predicate for some submodule over $\mathrm{End}(A)$.
Consider the expansion $T_S$ of a theory $T$ by a predicate for a submodel of a reduct $T_0$ of $T$. We present a setup in which this expansion admits a model companion $TS$. We show that the nice features of the theory $T$ transfer to $TS$. In particular, we study conditions for which this expansion preserves the $\NSOP{1}$-ness, the simplicity or the stability of the starting theory $T$. We give concrete examples of new $\NSOP{1}$ not simple theories obtained by this process, among them the expansion of a perfect $ω$-free $\PAC$ field of positive characteristic by generic additive subgroups, and the expansion of an algebraically closed field of \emph{any} characteristic by a generic multiplicative subgroup.
We study the generic theory of algebraically closed fields of fixed positive characteristic with a predicate for an additive subgroup, called $\mathrm{ACFG}$. This theory was introduced recently as a new example of $\mathrm{NSOP}_1$ non simple theory. In this paper we describe more features of $\mathrm{ACFG}$, such as imaginaries. We also study various independence relations in $\mathrm{ACFG}$, such as Kim-independence or forking independence, and describe interactions between them.
We consider the structure $(\mathbb{Z},+,0,|_{p_{1}},\dots,|_{p_{n}})$, where $x|_{p}y$ means $v_{p}(x)\leq v_{p}(y)$ and $v_p$ is the $p$-adic valuation. We prove that its theory has quantifier elimination in the language $\{+,-,0,1,(D_{m})_{m\geq1},|_{p_{1}},\dots,|_{p_{n}}\}$ where $D_m(x)\leftrightarrow \exists y ~ my = x$, and that it has dp-rank $n$. In addition, we prove that a first order structure with universe $\mathbb{Z}$ which is an expansion of $(\mathbb{Z},+,0)$ and a reduct of $(\mathbb{Z},+,0,|_{p})$ must be interdefinable with one of them. We also give an alternative proof for Conant's analogous result about $(\mathbb{Z},+,0,<)$.