Coarse Dimension and Pseudofinite Difference Fields

Transcription

1 Coarse Dimension and Pseudofinite Difference Fields Tingxiang Zou PhD student at Institut Carmille Jordan, Université Lyon 1 Supervisor: Frank Wagner April 9, 2018

2 Difference Fields A Difference Field is a field (F, +,, 0, 1) together with a field automorphism σ which is surjective (hence σ 1, σ 2, exist). L σ the language of difference rings is the language of rings augmented by a uniry function symbol σ. We fix an ambient difference field L. Let A be a subset, we denote by A σ to be the smallest difference subfield containing A and closed under σ and σ 1. Let E be a difference subfield and a be a tuple. The σ-degree, deg σ (a/e), is the transcendence degree of (E, a) σ over E. Let E be a difference subfield and a an element. We say a is transformally transcendental over E, if there is no non-zero difference polynomial over E vanishing on a. There is also the notion of transformal transcendence degree.

3 Examples and Motivation ACFA: (Chatzidakis, Hrushovski,...) The model companion of difference fields. Supersimple of SU-rank ω. For an element SU(a/A) < ω iff deg σ (a/a σ ) < iff a is not transformally transcendental over A σ. Let K q = (F alg p, Frob q ) where q is a power of the prime p. (Hrushovski) Any non-trivial ultraproduct along the class {K q q a power of some prime} is a model of ACFA. Finite σ-degree difference subfields of models of ACFA: (Ryten) Fix a prime p and m, n N coprime, then C p,m,n = {(F p kn+m, Frob p k ) k N} is one-dimensional asymptotic class. Any non-trivial ultraproduct of C p,m,n is a finite σ-degree difference subfield of k N K p k /U ACFA p. What can we say about pseudofinite difference fields with tranformally transcendental elements?

4 Coarse Dimension Let M = i I M i /U be a pseudofinite structure and X M be a pseudofinite subset.the coarse dimension with respect to X is a function: δ X : Def (M) R 0 {± } defined by δ X (Y ) = st.(log Y / log X ). We say δ X is continuous if for any ϕ(x, y) and α < β R, there is -definable D such that {y δ X (ϕ(m x, y)) α} D {y δ X (ϕ(m x, y)) < β}. For a tuple a and A M, define δ X (a/a) = inf δ X {ϕ ϕ tp(a/a)}. (Hrushovski) If δ X is continuous, then δ X is additive: δ X (a, b/a) = δ X (a/a, b) + δ X (b/a).

5 Main result Theorem There is a function f N N and let S = (F p kp, Frob p )/U k p f (p), U non-principal ultrafilter. p P For any (F, Frob) S, the pseudofinite coarse dimension with respect to F, δ F, of any L σ -definable set is integer-valued. Moreover, δ F is continuous in L σ. In particular, δ F is additive on definable sets (without expanding the language). Remark: The statement also holds for char(f ) = p > 0, i.e., for i I (F p k i, Frob p t i )/U provided k i >> t i for almost all i.

6 Proof Statement: Let (F, Frob) = p P (F p kp, Frob p )/U S. Then δ F is integer-valued. Fix an L σ -formula ϕ(x, a) with a = (a p ) p P /U For p P, translate ϕ(x, y) to ϕ p (x, y) L ring replacing t by t p. (Chatzidakis, van den Dries and Macintyre) ϕ p (x, a p ) has a dimension and measure (d p, µ p ) with d p x. ϕ p (F p kp, a p ) µ p (p kp ) dp ± C p (p kp ) dp 1/2 U will choose one d among all d p (as d p x ) log ϕ p(f lim p kp,ap) p U = d provided k log p kp p >> µ p, C p for almost all p.

7 Small fixed field δ F = 0 sets are wild! Let (F, Frob) S and Fix(F ) = {x F σ(x) = x} = p P F p /U. δ F (Fix(F )) = 0. (Folklore) If Char(F ) 2, there is an L ring -formula ϕ(x, y) such that for any pseudofintie subset A Fix(F ) there is b F ϕ(f, b) Fix(F ) = A. Corollary Th(F, Frob) has TP2 and strict order property. Algebraic closure There exist (F, Frob) S such that for any n, there are elements a n, b n with a n dcl Lσ (b n ) but deg σ (a n /(b n ) σ ) = n. Philosophy: Dimension 0 sets are uncontrollable, but everything should be good up to a dimension 0 error.

8 δ F and dim rk Goal: Relate δ F (a/a) to something we know. (F, Frob) p P (F alg p, Frob p )/U = ( F, Frob) ACFA. For a and A in ( F, Frob), we have SU(a/A) = ω k + n, define an integer-valued additive dimension dim rk (a/a) = k. dim rk (a/a) is the transformal transcendence degree of a over A σ. Let a and A in (F, Frob), then δ F (a/a) dim rk (a/a). Theorem Let ϕ(x) = yψ(x, y) be an L σ -formula such that ψ(x, y) is quantifier-free with parameters in a finite set A F. Then dim rk (a/a) δ F (ϕ(f x )) for any (F, Frob) ϕ(a). In particular, if (F, Frob) ϕ(a) and δ F (ϕ(f x )) = δ F (a/a), then dim rk (a/a) = δ F (a/a).

9 Proof and application ϕ(x, y) existential and y parameter. ONS if δ F (ϕ((f, Frob) x, b)) = 0 then deg σ (ϕ(x, b)/{b} σ ) <. If δ F (ϕ(x, b)) = 0 then ϕ p (F alg p, b p ) <. ϕ(x, b p ) has finitely many solutions in (F alg p, Frob p ) for almost all p. (Ryten, Tomašić) Estimate number of solutions in (F alg p, Frob p ) uniform in p. deg σ (ϕ(x, b)/{b} σ ) < d for some d. Corollary: Let (F, Frob) S and G be an existential-definable subgroup of an algebraic group H(F ).Then there is a quantifier-free definable group Ḡ G in F such that δ F (Ḡ) = δ F (G).

10 Not Yet Known δ F (a/a) = dim rk (a/a) for all a and A in F? For all definable group G there is some quantifier-free definable group Ḡ such that δ F (Ḡ) = δ F (G) = δ F (G Ḡ)? Is the common theory of (F, Frob) S recursively axiomatizable? Is there i I (F p k i, Frob p t i )/U with k i >> t i for almost all i, that is NTP2 or NSOP? Does the class of pseudofinite difference fields with transformally transcendental elements has a model companion?... Thank You!