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Model theory of exponentials on Lie algebras

Published online by Cambridge University Press:  01 February 2008

ANGUS MACINTYRE
ANGUS MACINTYRE*
Affiliation:
Department of Mathematical Sciences, Queen Mary University of London, Mile End, London E1 4NS, United Kingdom Email: angus@dcs.qmul.ac.uk
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Abstract

This paper presents an analysis of definitions and decidability for exponential functions on various matrix algebras. The main idea is to show that, generically, the entries of the exponential (or logarithm) of a matrix are Pfaffian functions of the entries of the matrix.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 18 , Special Issue 1: in memory of Sauro Tulipani , February 2008 , pp. 189 - 204
Copyright
Copyright © Cambridge University Press2008

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References

Barut, A. O., Zeni, J. R. and Laufer, A. (1994) The exponential map for the conformal group O(2,4). arXiv:hep-th/9408105v3.CrossRefGoogle Scholar
Bertolin, C. (2002) Périodes de 1-motifs et transcendance (in French). J. Number Theory 97 (2)204–221.CrossRefGoogle Scholar
Bianconi, R. (1990) Some Results in the Model Theory of Analytic Functions, Ph.D. Thesis, Oxford.Google Scholar
Bianconi, R. (1997) Nondefinability Results for Expansions of the Field of Real Numbers by the Exponential Function and by the Restricted Sine Function. J. Symb. Log. 62 (4)1173–1178.CrossRefGoogle Scholar
Gabrielov, A. and Vorobjov, N. (2004) Complexity of computations with Pfaffian and Noetherian functions. In: Normal forms, bifurcations and finiteness problems in differential equations. NATO Sci. Ser. II Math. Phys. Chem. 137 211–250.Google Scholar
Macintyre, A. J. (2005a) Elementary Theory of Elliptic Functions 1; The Formalism and a Special Case. In: Edmundo, M., Richardson, D. and Wilkie, A. (eds.) O-minimal Structures, Proceedings of the RAAG Summer School Lisbon 2003, Lecture Notes in Real Algebraic and Analytic Geometry, Cuvillier Verlag 104–131.Google Scholar
Macintyre, A. J. (2005b) Some Observations about the Real and Imaginary Parts of Complex Pfaffian Functions. To appear in: Macpherson, D. et al. . (eds.) Proceedings of INI 2005 Model Theory meeting.Google Scholar
Macintyre, A. J. and Wilkie, A. (1996) On the decidability of the real exponential field. In: Odifreddi, P. (ed.) Kreiseliana. About and Around Georg Kreisel, A. K. Peters 441–467.Google Scholar
Serre, J. P. (1965) Lie Algebras and Lie groups, W. A. Benjamin.Google Scholar
Wilkie, A. (1996) Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J Amer Math Soc 9 (4)1051–1094.CrossRefGoogle Scholar
Zilber, B. I. (2004) Pseudo-exponentiation on algebraically closed fields of characteristic zero. Ann. Pure Appl. Logic 132 (1)67–95.CrossRefGoogle Scholar