Quantifier elimination for algebraic $D$-groups

Abstract

We prove that if $G$ is an algebraic $D$-group (in the sense of Buium over a differentially closed field $(K,\partial )$ of characteristic $0$, then the first order structure consisting of $G$ together with the algebraic $D$-subvarieties of $G, G\times G,\dots$, has quantifier-elimination. In other words, the projection on $G^{n}$ of a $D$-constructible subset of $G^{n+1}$ is $D$-constructible. Among the consequences is that any finite-dimensional differential algebraic group is interpretable in an algebraically closed field.