Seminários de Teoria dos Modelos

25 de Março 2009, 16h30

Some minimal sets in ACFA
Complexo Interdisciplinar, Sala B3-01

ALICE MEDVEDEV (University of Illinois at Chicago, U.S.A)

Abstract:

The theory of difference fields (i.e. fields with a distinguished automorphism $\sigma$) admits a model-companion ACFA. This is a supersimple theory where finite U-rank types can be analyzed in terms of "minimal" (U-rank 1) types, and the minimal types satisfy the Zilber trichotomy.

The finite U-rank definable sets in ACFA come up in arithmetic geometry and algebraic dynamics, for example in Hrushovski's work on the Mamin-Mumford conjecture, where $\sigma$-varieties (sets defined by $\sigma(x)=f(x)$) are the central object. When $x$ is a single variable and $f$ is a rational function, this set is minimal. My thesis shows that these sets are uniform in the Zilber trichotomy, and characterizes the three cases.

My recent work with Thomas Scanlon on rational dynamics further clarifies the properties of these sets in characteristic 0 when $f$ is a polynomial, characterizing nonorthogonality between them and the algebraic closure on them, in particular showing that the trivial ones have finite Morley rank.

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