Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-13T22:37:22.516Z Has data issue: false hasContentIssue false

On subgroups of the additive group in differentially closed fields

Published online by Cambridge University Press:  12 March 2014

Sonat Süer*
Affiliation:
Department of Mathematics, Istanbul Bilgi University, Kurtulus Deresi Caddesi No: 47, 34440 Dolapdere, Turkey, E-mail: [email protected]

Abstract

In this paper we deal with the model theory of differentially closed fields of characteristic zero with finitely many commuting derivations. First we observe that the only known lower bound for the Lascar rank of types in differentially closed fields, announced in a paper of McGrail, is false. This gives us a new class of regular types which are orthogonal to fields. Then we classify the subgroups of the additive group of Lascar rank omega with differential-type 1 which are nonorthogonal to fields. The last parts consist of an analysis of the quotients of the heat variety. We show that the generic type of such a quotient is locally modular. Finally, we answer a question of Phylliss Cassidy about the existence of certain Jordan-Hölder type series in the negative.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2012

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

[1] Cartan, Elie, Les groupes de transformations continus, infinis, simples. Annales Scientifiques de l'École Normale Supérieure (3), vol. 26 (1909), pp. 93161.CrossRefGoogle Scholar
[2] Cassidy, Phyllis Joan, Differential algebraic groups., American Journal of Mathematics, vol. 94 (1972), pp. 891954.CrossRefGoogle Scholar
[3] Cassidy, Phyllis Joan, The differential rational representation algebra on a linear differential algebraic group, Journal of Algebra, vol. 37 (1975), pp. 223238.CrossRefGoogle Scholar
[4] Cassidy, Phyllis Joan, Unipotent differential algebraic groups. Contributions to algebra (collection of papers dedicated to Ellis Kolchin), Société Mathématique de France, 1978, pp. 83115.Google Scholar
[5] Cassidy, Phyllis Joan, The classification of the semisimple differential algebraic groups and the linear semisimple differential algebraic lie algebras., Journal of Algebra, vol. 121 (1989), pp. 169238.Google Scholar
[6] Kolchin, Ellis, Differential algebra and algebraic groups, Academic Press, 1973.Google Scholar
[7] Kolchin, Ellis, Differential algebraic groups, Academic Press, 1985.Google Scholar
[8] Marker, David, Introduction to the model theory of fields, Model theory of fields, Lecture Notes in Logic, vol. 5, Association for Symbolic Logic, A K Peters, Ltd., 2006.Google Scholar
[9] Mcgrail, Tracey, The model theory of differentialfields with finitely many commuting derivations, this Journal, vol. 65 (2000), pp. 885913.Google Scholar
[10] Moosa, Rahim, Pillay, Anand, and Scanlon, Thomas, Differential arcs and regular types in differential fields, Journal fur die reine und angewandte Mathematik, vol. 620 (2008), pp. 3554.Google Scholar
[11] Pillay, Anand, Geometric stability theory, Oxford Logic Guides, vol. 32, Oxford University Press, 1996.Google Scholar
[12] Pillay, Anand, Some foundational questions concerning differential algebraic groups., Pacific Journal of Mathematics, vol. 179 (1997), pp. 179200.CrossRefGoogle Scholar
[13] Pillay, Anand and Yan Pong, Wai, On Lascar rank and Morley rank of definable groups in differentially closedfields., this Journal, vol. 67 (2002), pp. 11891196.Google Scholar
[14] Pillay, Anand and Sokolović, Željko, A remark on differential algebraic groups, Communications in Algebra, vol. 20 (1992), pp. 30153026.CrossRefGoogle Scholar
[15] Poizat, Bruno, Stable groups, Mathematical Surveys and Monographs, vol. 87, American Mathematical Society, 2001.Google Scholar
[16] Yan Pong, Wai, Some applications of ordinal dimensions to the theory of differentially closed fields, this Journal, vol. 65 (2000), pp. 347376.Google Scholar
[17] Yan Pong, Wai, Rank inequalities in the theory of differentially closedfields, Logic colloquium '03 (Stoltenberg-Hansen, Viggo and Vaananen, Jouko, editors). Lecture Notes in Logic, vol. 24, Association for Symbolic Logic, 2006, pp. 232243.CrossRefGoogle Scholar
[18] Robinson, Abraham, On the concept of a differentially closedfield, Bulletin of the Research Council of Israel Section F, vol. 8F (1959), pp. 113128.Google Scholar
[19] Yu Sit, William, Typical differential dimension of the intersection of linear differential algebraic groups., Journal of Algebra, vol. 32 (1974), pp. 627698.Google Scholar
[20] Yu Sit, William, Differential algebraic subgroups of SL(2) and strong normality in simple extensions, American Journal of Mathematics, vol. 97 (1975), pp. 627698.CrossRefGoogle Scholar
[21] Suer, Sonat, Model theory of differentially closed fields with several commuting derivations, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2007.Google Scholar
[22] Van Der Waerden, B. L., Moderne Algebra, vol. II, Springer Verlag, 1931.Google Scholar