A criterion for uniform finiteness in the imaginary sorts

Let T be a theory. If T eliminates \(\exists ^\infty \), it need not follow that \(T^{\mathrm {eq}}\) eliminates \(\exists ^\infty \), as shown by the example of the p-adics. We give a criterion to determine whether \(T^{\mathrm {eq}}\) eliminates \(\exists ^\infty \). Specifically, we show that \(T^{\mathrm {eq}}\) eliminates \(\exists ^\infty \) if and only if \(\exists ^\infty \) is eliminated on all interpretable sets of “unary imaginaries.” This criterion can be applied in cases where a full description of \(T^{\mathrm {eq}}\) is unknown. As an application, we show that \(T^{\mathrm {eq}}\) eliminates \(\exists ^\infty \) when T is a C-minimal expansion of ACVF.