Fields with operators

Some documents

• “Geometric characterizations of existentially closed fields with operators”, Illinois Journal of Mathematics Volume 48, Issue 4 (Winter 2004), pp. 1321–1343.
• Slides (and older versions, etc.)
• “Differential forms in the model theory of differential fields”, Journal of Symbolic Logic 68 (3) 2003, pp. 923–945.
• “A Note on the Axioms for Differentially Closed Fields of Characteristic Zero”. With Anand Pillay. Journal of Algebra 204, 108-115 (1998).
• “Differential fields”: from the course by Anand Pillay given at Fields Institute, 1996; these notes were typed and edited by me and other students, who are not credited by name in: Lectures on algebraic model theory (Bradd Hart and Matthew Valeriote, editors; Fields Institute monographs 15, 2002).

Commentary:

“Geometric characterizations of existentially closed fields with operators”

Contents of the paper:

• Two new geometric axiomatizations for DCF (arbitrary characteristic) are given.
• One of these generalizes to DCF^m.
• The other is parallel to an alternative geometric axiomatization of ACFA.
• The model-companion of the theory of fields with a derivation or an endomorphism can be axiomatized in a uniform way (without distinguishing the two cases).
• The theory of fields with a derivation and an endomorphism is not companionable in positive characteristic.

Cited by:

1. Kowalski, Piotr. Derivations of the Frobenius map. J. Symbolic Logic 70 (2005), no. 1, 99--110. MR2119125 (2005m:03069)
2. Kowalski, Piotr. Geometric axioms for existentially closed Hasse fields. Ann. Pure Appl. Logic 135 (2005), no. 1-3, 286--302. MR2156140 (2006d:03061)
3. Pillay, Anand; Polkowska, Dominika. On PAC and bounded substructures of a stable structure. J. Symbolic Logic 71 (2006), no. 2, 460--472. MR2225887

“Differential forms in the model theory of differential fields”

In this paper, a differential field can be understood as a pair (K,V), where K is a field of characteristic zero, and V is a finite-dimensional space of derivations from K to itself that is closed under the Lie bracket. A first-order structure is determined by choice of a spanning set for V, and the definable sets are independent of this choice. The existentially closed such structures are characterized by a kind of generalization of the Frobenius Theorem of differential geometry. The characterization is first-order; in particular, new axioms for the theory fields of characteristic zero with m commuting derivations can be given.

Cited by:

1. Scanlon, Thomas. Model theory and differential algebra. Differential algebra and related topics (Newark, NJ, 2000), 125--150, World Sci. Publishing, River Edge, NJ, 2002. MR1921697 (2003g:03062)
2. Pierce, David. Geometric characterizations of existentially closed fields with operators. Illinois J. Math. 48 (2004), no. 4, 1321--1343. MR2114160 (2006e:03053)
3. Tressl, Marcus. The uniform companion for large differential fields of characteristic 0. Trans. Amer. Math. Soc. 357 (2005), no. 10, 3933--3951 (electronic). MR2159694

“A Note on the Axioms for Differentially Closed Fields of Characteristic Zero”

In this paper, so-called geometric axioms for the theory of fields of characteristic zero with one derivation. They can be seen as a special case of the axioms in “Differential forms in the model theory of differential fields”.

The paper is cited by:

1. Marker, David. Model theory of differential fields. Model theory, algebra, and geometry, 53--63, Math. Sci. Res. Inst. Publ., 39, Cambridge Univ. Press, Cambridge, 2000. MR1773702 (2001j:12005)
2. Hrushovski, Ehud; Pillay, Anand. Effective bounds for the number of transcendental points on subvarieties of semi-abelian varieties. Amer. J. Math. 122 (2000), no. 3, 439--450. MR1759883 (2001d:11078)
3. Scanlon, Thomas. Model theory and differential algebra. Differential algebra and related topics (Newark, NJ, 2000), 125--150, World Sci. Publishing, River Edge, NJ, 2002. MR1921697 (2003g:03062)
4. Pierce, David. Differential forms in the model theory of differential fields. J. Symbolic Logic 68 (2003), no. 3, 923--945. MR2000487 (2004h:03080)
5. Pillay, Anand; Ziegler, Martin. Jet spaces of varieties over differential and difference fields. Selecta Math. (N.S.) 9 (2003), no. 4, 579--599. MR2031753 (2004m:12011)
6. Pillay, Anand. Algebraic $D$-groups and differential Galois theory. Pacific J. Math. 216 (2004), no. 2, 343--360. MR2094550 (2005k:12007)
7. Pierce, David. Geometric characterizations of existentially closed fields with operators. Illinois J. Math. 48 (2004), no. 4, 1321--1343. MR2114160 (2006e:03053)
8. Kowalski, Piotr. Derivations of the Frobenius map. J. Symbolic Logic 70 (2005), no. 1, 99--110. MR2119125 (2005m:03069)
9. Kowalski, Piotr. Geometric axioms for existentially closed Hasse fields. Ann. Pure Appl. Logic 135 (2005), no. 1-3, 286--302. MR2156140 (2006d:03061)
10. Tressl, Marcus. The uniform companion for large differential fields of characteristic 0. Trans. Amer. Math. Soc. 357 (2005), no. 10, 3933--3951 (electronic). MR2159694
11. Michaux, Christian; Rivière, Cédric. Quelques remarques concernant la théorie des corps ordonnés différentiellement clos. (French) [The theory of ordered differentially closed fields] Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 3, 341--348. MR2173697
12. Guzy, Nicolas. Note sur les corps différentiellement clos valués. (French) [Note on differentially closed valued fields] C. R. Math. Acad. Sci. Paris 341 (2005), no. 10, 593--596. MR2179796 (2006e:03052)
13. Kowalski, Piotr; Pillay, Anand. Quantifier elimination for algebraic $D$-groups. Trans. Amer. Math. Soc. 358 (2006), no. 1, 167--181 (electronic). MR2171228 (2006i:03051)
14. Pillay, Anand; Polkowska, Dominika. On PAC and bounded substructures of a stable structure. J. Symbolic Logic 71 (2006), no. 2, 460--472. MR2225887