Geometric axioms for differentially closed fields with several commuting derivations

Sánchez, Omar León

doi:10.1016/j.jalgebra.2012.03.043

Cited by 9 publications

(11 citation statements)

References 15 publications

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“…In this section, we will show that ∆-Chow varieties of index (n, d, s, r) exist for all chosen n, d, s, r. Similar to the ordinary differential case, the main idea is to first definably embed G (n,d,s,r) into a finite disjoint union C of the chosen algebraic Chow varieties and then show the image of G (n,d,s,r) is a definable subset of C. So, the language from model theory of partial differentially closed fields (see [21,24,27]) will be used and we assume E is a ∆-closed field of characteristic 0 (i.e., E |= DCF 0,m ) throughout this section.…”

Section: The Existence Of a Type Of Partial Differential Chow Varietiesmentioning

confidence: 99%

Partial differential Chow forms and a type of partial differential Chow varieties

2020

Communications in Algebra

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We first present an intersection theory of partial differential varieties with quasi-generic differential hypersurfaces. Then, based on the generic differential intersection theory, we define the partial differential Chow form for an irreducible partial differential variety V of Kolchin polynomial ω V (t) = (d + 1) t+m m − t+m−s m . And we establish for the partial differential Chow form most of the basic properties of the ordinary differential Chow form. Furthermore, we prove the existence of a type of partial differential Chow varieties.

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Section: The Existence Of a Type Of Partial Differential Chow Varietiesmentioning

confidence: 99%

Partial differential Chow forms and a type of partial differential Chow varieties

2020

Communications in Algebra

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“…The notion of relative prolongation of an affine ∆-algebraic variety was introduced in [9], where it was used to formulate geometric first-order axioms for partial differentially closed fields.…”

Section: Is a Derivation On K Commuting With ∆)mentioning

confidence: 99%

Relative D-groups and differential Galois theory in several derivations

Sánchez

2015

Trans. Amer. Math. Soc.

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“…For many geometrically well behaved theories one can prove the existence of a model companion by adapting the axiomatization of ACF A given by Chatzidakis and Hrushovski in terms of algebro-geometric objects [7]. These so called "geometric axiomatizations" have succesfully been applied to yield the existence of model companions in several interesting theories: ordinary differential fields [29], partial differential fields (several commuting derivations) [21], [22], fields with commuting Hasse-Schmidt derivations in positive characteristic [20], fields with free operators [25], and in theories having a "geometric notion of genericity" [15].…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by the methods in [22], we bypass the above definability issues by applying the differential algebraic machinery of characteristic sets of prime differential ideals developed by Kolchin [18] (and Rosenfeld [35]). More precisely, we prove a characterization of the existentially closed models in terms of characteristic sets of prime differential ideals (see Theorem 2.3).…”

Section: Introductionmentioning

confidence: 99%