Existential rank and essential dimension of diophantine sets
Nicolas Daans, Philip Dittmann, Arno Fehm
TL;DR
The paper introduces and studies three quantitative invariants for diophantine sets over fields—the existential rank ${\rm rk}^{\exists}$, the positive-existential rank ${\rm rk}^{\exists^{+}}$, and the essential fibre dimension ${\rm efd}$—and connects them to essential dimension via a geometric framework. It develops model-theoretic criteria and inequalities, relates these ranks to the theories of fields, and shows how ${\rm efd}$ governs the fibre-wise complexity of existential definitions, with ${\rm efd}$ typically smaller than ranks and often zero in canonical theories (ACF, RCF, pCF_d). The paper further ties these invariants to canonical dimension and incompressible varieties to derive lower bounds, and provides a lifting technique to transfer lower bounds to complete theories of fields, producing concrete results for quadrics, cyclic norms, and related objects. Finally, it analyzes global fields, proving a universal lower bound ${\rm rk}^{\exists}(K) \ge 2$ and discussing the implications for diophantine definability and the existence of universal diophantine sets, while highlighting open problems regarding $\mathbb{Q}$ and $\mathbb{F}_p(t)$.
Abstract
We study the minimal number of existential quantifiers needed to define a diophantine set over a field and relate this number to the essential dimension of the functor of points associated to such a definition.