Jet and prolongation spaces

Moosa, Rahim; Scanlon, Thomas

doi:10.1017/s1474748010000010

Cited by 30 publications

(105 citation statements)

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“…Although fields with several commuting automorphisms and fields with several derivations and automorphisms commuting with each other have been studied from the standpoints of algebra [11,22], model theory [6,7,21], and symbolic computation [15,23], we are not aware of any analogues of the Primitive Element Theorem for such fields. Another common generalization of fields equipped with a derivations and fields equipped with an automorphism is the theory of fields with free operators introduced in [27,28] (see also [4,17]). We are not aware of any analogues of the Primitive Element Theorem for such fields.…”

Section: Overview and Prior Resultsmentioning

confidence: 99%

A primitive element theorem for fields with commuting derivations and automorphisms

Pogudin

2019

Sel. Math. New Ser.

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We establish a Primitive Element Theorem for fields equipped with several commuting operators such that each of the operators is either a derivation or an automorphism. More precisely, we show that for every extension F ⊂ E of such fields of zero characteristic such that • E is generated over F by finitely many elements using the field operations and the operators,• every element of E satisfies a nontrivial equation with coefficient in F involving the field operations and the operators,• the action of the operators on E is irredundant there exists an element a ∈ E such that E is generated over F by a using the field operations and the operators. This result generalizes the Primitive Element Theorems by Kolchin and Cohn in two directions simultaneously: we allow any numbers of derivations and automorphisms and do not impose any restrictions on the base field F .

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Section: Overview and Prior Resultsmentioning

confidence: 99%

A primitive element theorem for fields with commuting derivations and automorphisms

Pogudin

2019

Sel. Math. New Ser.

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“…is the graph of a C-linear isomorphism from Jet n (X s1 ) x1 to Jet n (X s2 ) x2 . (For this latter property of jet spaces, see for example Lemma 5.10 of [28], which works in the algebraic setting but goes through in our analytic setting.) As this is a definable property we get for c ∈ (F, G) ♯ , with c generic in F over B, that Jet n (G) (c,σ(c)) is the graph of a (CCM-definable) K-linear isomorphism g : Jet n (F ) c → Jet n (F σ ) σ(c) .…”

Section: Difference-analytic Jet Spaces and The Zilber Dichotomymentioning

confidence: 96%

Model theory of compact complex manifolds with an automorphism

Bays

Hils

Moosa

2017

Trans. Amer. Math. Soc.

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Motivated by possible applications to meromorphic dynamics, and generalising known properties of difference-closed fields, this paper studies the theory CCMA of compact complex manifolds with a generic automorphism. It is shown that while CCMA does admit geometric elimination of imaginaries, it cannot eliminate imaginaries outright: a counterexample to 3-uniqueness in CCM is exhibited. Finite-dimensional types are investigated and it is shown, following the approach of Pillay and Ziegler, that the canonical base property holds in CCMA. As a consequence the Zilber dichotomy is deduced: finitedimensional minimal types are either one-based or almost internal to the fixed field. In addition, a general criterion for stable embeddedness in T A (when it exists) is established, and used to determine the full induced structure of CCMA on projective varieties, simple nonalgebraic complex tori, and simply connected nonalgebraic strongly minimal manifolds.

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“…, ℓ. n of the set of equations {f e (x) = 0 : f ∈ I ∆ (V /K)}. (c) In the pure D-fields case when ∆ = ∅, the prolongations introduced here coincide with the D-prolongations introduced in [6] and used in [7]. (d) When D(R) = R × R, the prolongation introduced here is the differencedifferential prolongation V × V σ that was used in [5].…”

Section: D-prolongations Of ∆-Varietiesmentioning

confidence: 97%

“…Taking our cue from the general theory of abstract prolongations of algebraic varieties developed in [6], the construction should look something like this: base change V to D(K) via the homomorphism e : K → D(K) and then take the Weil restriction back down to K via the standard K-algebra structure on D(K) = K ⊗ k B. Since the theory of Weil restrictions is not to our knowledge developed in the differential context, we will instead give this construction explicitly in co-ordinates.…”

Section: D-prolongations Of ∆-Varietiesmentioning

confidence: 99%