Alice Medvedev (UIC Math)
Some tractable trivial minimal sets in ACFA
Abstract: Let $(K, \sigma)$ be a model of ACFA of characteristic $0$, and let $f(x)$ be a polynomial over $K$. We completely describe the definable structure on the Lascar-rank $1$ set $f^\sharp$ defined by $\sigma(x)= f(x)$. Chatzidakis and Hrushovski proved the Zilber Trichotomy conjecture in ACFA, and characterized the functions that give fieldlike $f^\sharp$. Since all of these are isomorphic to the fixed field of $\sigma$, it is easy to see that the definable structure on them is terribly complicated. My thesis characterized the functions that give grouplike $f^\sharp$, and Chatzidakis, Hrushovski, and others showed that the corresponding groups are one-based. We now complete the picture by describing the quantifier-free definable relations on $f^\sharp$, which pins down the algebraic closure operator; the (lack of) quantifierfully definable relations on $f^\sharp$, which shows that most of them are strongly minimal; and the non-orthogonality relation between different such sets, which turns out to be occasionally definable and otherwise describable. |
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