Generic multiplicative endomorphism of a field

TL;DR

The paper introduces the model companion $ACFH$ for fields expanded by a multiplicative endomorphism, establishes its NSOP$_1$-nature and non-simplicity, and develops a geometric, first-order framework for existential closure via $m$-varieties and multiplicative freeness. It provides a complete axiomatisation and model-completeness, including a reduction to affine curves and a detailed analysis of completions, types, and kernels. The study reveals that kernels of polynomial endomorphisms $\ker P(\theta)$ are generic and pseudofinite-cyclic, with iterations $\theta^{(n)}$ preserving genericity; these results place $ACFH$ as a robust template for generic expansions with intricate independence phenomena. The work further shows that elimination of imaginaries holds under the existence axiom for forking, and it lays groundwork for future exploration of generic endomorphisms in algebraic and model-theoretic contexts, including potential extensions to other groups and varieties.

Abstract

We introduce the model-companion of the theory of fields expanded by a unary function for a multiplicative map, which we call ACFH. Among others, we prove that this theory is NSOP$_1$ and not simple, that the kernel of the map is a generic pseudo-finite abelian group. We also prove that if forking satisfies existence, then ACFH has elimination of imaginaries.

Theorems & Definitions (121)

• Theorem A
• Theorem B
• Theorem C
• Lemma 1.1
• proof
• Lemma 1.2
• proof
• Remark 1.3
• Remark 1.4: Full Existence
• Lemma 1.5: Amalgamation Property