Differential fields

David Pierce

In a differential field, how can we tell whether all consistent systems of equations and inequations have solutions? I shall review the history of answers to this question, and I shall update the accounts in [P1, P2]. To begin with the Robinsonian beginnings, I remind or inform the reader-listener of the following. The class of substructures of models of a theory T is elementary, and its theory is T_{∀}. The class of structures in which a structure 𝔐 embeds is elementary, and its theory is diag(𝔐). The class of models of T is closed under unions of chains if and only if T = T_{∀∃} [R2, 3.4.7]. The theory T is called model-complete [R1] if T ∪ diag(𝔐) is complete whenever 𝔐 ⊧ T. If T ⊆ T^{∗}, and T_{∀} = T^{∗}_{∀}, then T^{∗} is the model-completion [R2] of T if T^{∗} ∪ diag(𝔐) is complete whenever 𝔐 ⊧ T; but T^{∗} is merely the model-companion of T if T^{∗} is model-complete. A derivation of a field K is an additive endormorphism D of K that respects the Leibniz rule, D(x · y) = Dx · y + x · Dy. A differential field is a field equipped with one or more derivations.

References

[R1] Abraham Robinson. Complete theories. 1956.

[R2] Abraham Robinson. Introduction to model theory. 1963.

[W1] Carol Wood. The model theory of diff. fields of characteristic p ≠ 0. 1973.

[W2] Carol Wood. Prime model extensions for diff. fields of char. p ≠ 0. 1974.

[B] Lenore Blum. Differentially closed fields: a model-theoretic tour. 1977.

[S] Michael F. Singer. The model theory of ordered differential fields. 1978.

[PP] D. Pierce and Anand Pillay. A note on the axioms for differentially closed fields of characteristic zero. 1998.

[McG] Tracey McGrail. The model theory of differential fields with finitely many commuting derivations. 2000.

[Y] Yoav Yaffe. Model completion of Lie differential fields. 2001.

[HI] E. Hrushovski and M. Itai. On model complete differential fields. 2003.

[P1] D. Pierce. Diff. forms in the model theory of diff. fields. 2003.

[P2] D. Pierce. Geometric characterizations of e.c. fields with operators. 2004.

[K] Piotr Kowalski. Derivations of the Frobenius map. J. Symbolic Logic, 70(1):99– 110, 2005.

[T] Marcus Tressl. The uniform companion for large diff. fields of char. 0. 2005.

[MR] Christian Michaux and Cédric Rivière. Quelques remarques concernant la théorie des corps ordonnés différentiellement clos. 2005.

[GR] Nicolas Guzy and Cédric Rivière. Principle of differential lifting for theories of differential fields and Pierce–Pillay axiomatization. 20xx.

Orta Doğu Teknik Üniversitesi, Ankara

email: dpierce@metu.edu.tr
web: www.math.metu.edu.tr/^{∼}dpierce