Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-06T09:54:52.908Z Has data issue: false hasContentIssue false
  • English
  • Français

DIFFERENTIAL-ALGEBRAIC JET SPACES PRESERVE INTERNALITY TO THE CONSTANTS

Published online by Cambridge University Press:  22 July 2015

ZOE CHATZIDAKIS ,
MATTHEW HARRISON-TRAINOR  and
RAHIM MOOSA
ZOE CHATZIDAKIS
Affiliation:
DÉPARTEMENT DE MATHÉMATIQUES ET APPLICATIONS (UMR 8553) ECOLE NORMALE SUPÉRIEURE 45 RUE D’ULM, 75230 PARIS CEDEX 5FRANCEE-mail: [email protected]
MATTHEW HARRISON-TRAINOR
Affiliation:
UNIVERSITY OF CALIFORNIA BERKELEY, DEPARTMENT OF MATHEMATICS 970 EVANS HALL, BERKELEY, CA 94720-3840, USAE-mail: [email protected]
RAHIM MOOSA
Affiliation:
UNIVERSITY OF WATERLOO DEPARTMENT OF PURE MATHEMATICS 200 UNIVERSITY AVENUE WEST WATERLOO, ONTARIO N2L 3G1 CANADAE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Suppose p is the generic type of a differential-algebraic jet space to a finite dimensional differential-algebraic variety at a generic point. It is shown that p satisfies a certain strengthening of almost internality to the constants. This strengthening, which was originally called “being Moishezon to the constants” in [9] but is here renamed preserving internality to the constants, is a model-theoretic abstraction of the generic behaviour of jet spaces in complex-analytic geometry. An example is given showing that only a generic analogue holds in the differential-algebraic case: there is a finite dimensional differential-algebraic variety X with a subvariety Z that is internal to the constants, such that the restriction of the differential-algebraic tangent bundle of X to Z is not almost internal to the constants.

Keywords

differentially closed fieldsfinite rank definable setsinternality to the constantsKolchin’s differential tangent space
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 3 , September 2015 , pp. 1022 - 1034
Copyright
Copyright © The Association for Symbolic Logic 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Buium, A., Differential Algebraic Groups of Finite Dimension, vol. 1506, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1992.CrossRefGoogle Scholar
Cassidy, P., Differential algebraic groups. American Journal of Mathematics, vol. 94 (1972), pp. 891–954.CrossRefGoogle Scholar
Chatzidakis, Z., A note on canonical bases and one-based types in supersimple theories. Confluentes Mathematici, vol. 4 (2012), no. 3, 1250004.CrossRefGoogle Scholar
Kolchin, E. R., Differential algebraic groups, Pure and Applied Mathematics, vol. 114, Academic Press, Orlando, FL, 1985.Google Scholar
León Sánchez, O., Contributions to the Model Theory of Partial Differential Fields, Ph.D thesis, University of Waterloo, Waterloo, 2013.Google Scholar
León Sánchez, O., Relative D-groups and differential Galois theory in several derivations. Transactions of the American Mathematical Society, to appear.Google Scholar
Marker, D., Model theory of differential fields, Model Theory of Fields (Pillay, A.Marker, D., Messmer, M., editor), Lecture notes in logic, vol. 5, Springer, Berlin, 1996, pp. 38–113.CrossRefGoogle Scholar
Moosa, R., Jet spaces in complex analytic geometry: An exposition, e-print available athttp://arxiv.org/abs/math.LO/0405563, 2003.Google Scholar
Moosa, R., A model-theoretic counterpart to Moishezon morphisms, Models, Logics, and Higher-dimensional categories, CRM Proceedings & Lecture Notes, vol. 53, American Mathematical Society, Providence, RI, 2011, pp. 177–188.CrossRefGoogle Scholar
Moosa, R. and Pillay, A., On canonical bases and internality criteria. Illinois Journal of Mathematics, vol. 52 (2008), no. 3, pp. 901–917.CrossRefGoogle Scholar
Moosa, R. and Scanlon, T., Jet and prolongation spaces. Journal de l’Institut de Mathématiques de Jussieu, vol. 9 (2010), no. 2, pp. 391–430.Google Scholar
Moosa, R., Generalized Hasse-Schmidt varieties and their jet spaces. Proceedings of the London Mathematical Society. Third Series, vol. 103 (2011), no. 2, pp. 197–234.CrossRefGoogle Scholar
Pillay, A., Remarks on algebraic D-varieties and the model theory of differential fields, Logic in Tehran, Lecture notes in Logic, vol. 26, Association of Symbolic Logic, La Jolla, CA, 2006, pp. 256–269.Google Scholar
Pillay, A. and Ziegler, M., Jet spaces of varieties over differential and difference fields. Selecta Mathematica (N.S.), vol. 9 (2003), no. 4, pp. 579–599.CrossRefGoogle Scholar