We study algebraic dynamical systems (and, more generally, σ-varieties) Φ : A n C → A n C given by coordinatewise univariate polynomials by refining an old theorem of Ritt on compositional identities amongst polynomials. More precisely, we find a nearly canonical way to write a polynomial as a composition of "clusters" from which one may easily read off possible compositional identities. Our main result is an explicit description of the (weakly) skew-invariant varieties, that is, for a fixed field automorphism σ : C → C those algebraic varieties X ⊆ A n C for which Φ(X) ⊆ X σ . As a special case, we show that if f (x) ∈ C[x] is a polynomial of degree at least two which is not conjugate to a monomial, Chebyshev polynomial or a negative Chebyshev polynomial, and X ⊆ A 2 C is an irreducible curve which is invariant under the action of (x, y) → (f (x), f (y)) and projects dominantly in both directions, then X must be the graph of a polynomial which commutes with f under composition. As consequences, we deduce a variant of a conjecture of Zhang on the existence of rational points with Zariski dense forward orbits and a strong form of the dynamical ManinMumford conjecture for liftings of the Frobenius.We also show that in models of ACFA0, a disintegrated set defined by σ(x) = f (x) for a polynomial f has Morley rank one and is usually strongly minimal, that model theoretic algebraic closure is a locally finite closure operator on the nonalgebraic points of this set unless the skewconjugacy class of f is defined over a fixed field of a power of σ, and that nonorthogonality between two such sets is definable in families if the skewconjugacy class of f is defined over a fixed field of a power of σ.
Generalising and unifying the known theorems for difference and differential fields, it is shown that for every finite free algebra scheme D over a field A of characteristic zero, the theory of D-fields has a model companion D -CF 0 which is simple and satisfies the Zilber dichotomy for finite-dimensional minimal types.Contents the multiplication of D(R) may be expressed by insisting that certain polynomial relations hold amongst ∂ 0 (x), . . . , ∂ n−1 (x); ∂ 0 (y), . . . , ∂ n−1 (y); ∂ 0 (xy), . . . , ∂ n−1 (xy). In this way, the class of D-fields is easily seen to be first order in the language of rings augmented by unary function symbols for the operators ∂ 0 , . . . , ∂ n−1 . On the other hand, the interpretation of the operators as components of a ring homomorphism permits us to apply ideas from commutative algebra and algebraic geometry to analyze these theories.Our first main theorem is that for any ring scheme D (meeting the requirements set out in Section 3), the theory of D-fields of characteristic zero has a model companion, which we denote by D -CF 0 and call the theory of D-closed fields. Our axiomatization of D -CF 0 follows the geometric style which first appeared in the Chatzidakis-Hrushovski axioms for ACFA [3] and was then extended to differential fields by Pierce and Pillay [14]. Moreover, the proofs will be familiar to anyone who has worked through the corresponding results for difference and differential fields. Following the known proofs for difference and differential fields, we establish a quantifier simplification theorem and show that D -CF 0 is always simple.As noted above, the theory DCF 0 is the quintessential ω-stable theory, but ACFA, the model companion of the theory of difference fields is not even stable. At a technical level, the instability of ACFA may be traced to the failure of quantifier elimination which, algebraically, is due to the non-uniqueness (up to isomorphism) of the extension of an automorphism of a field to an automorphism of its algebraic closure. We show that this phenomenon, namely that instability is tied to the nonuniqueness of extensions of automorphisms, pervades the theory of D-fields. That is, for each D there is a finite list of associated endomorphisms expressible as linear combinations of the basic operators. Since we require that ∂ 0 is the identity map, one of these associated endomorphisms is always the identity map. If there are any others, then the theory of D -CF 0 suffers from instability and the failure of quantifier elimination just as does ACFA. On the other hand, if there are no other associated endomorphisms, then D -CF 0 is stable.The deepest of the fine structural theorems for types in DCF 0 and in ACFA is the Zilber dichotomy for minimal types, first established by Sokolović and Hrushovski for DCF 0 using Zariski geometries [8], for ACFA 0 by Chatzidakis and Hrushovski through a study of ramification [3], and by Chatzidakis, Hrushovski and Peterzil for ACFA in all characteristics using the theory of limit types and a refined form of the ...
Motivated by the problem of determining the structure of integral points on subvarieties of semiabelian varieties defined over finite fields, we prove a quantifier elimination result for certain modules over finite simple extensions of the integers given together with predicates for orbits of the distinguished generator of the ring.Remark 2.2. For fixed R = Z[F ], the following observations are immediate consequences of the definitions:• The class of F -spaces is closed under taking products and passing to quotients by F -subspaces.
The notion of a prolongation of an algebraic variety is developed in an abstract setting that generalizes the difference and (Hasse) differential contexts. An interpolating map that compares these prolongation spaces with algebraic jet spaces is introduced and studied.
The notion of a D-ring, generalizing that of a differential or a differenee ring, is introduced, Quantifier elimination and a version of the Ax—Kochen—Eršov principle is proven for a theory of valued D-fields of residual characteristic zero.
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