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Asymptotic Differential Algebra and Model Theory of Transseries$
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Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven

Print publication date: 2017

Print ISBN-13: 9780691175423

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691175423.001.0001

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Eventual Quantities, Immediate Extensions, and Special Cuts

Eventual Quantities, Immediate Extensions, and Special Cuts

Chapter:
(p.474) Chapter Eleven Eventual Quantities, Immediate Extensions, and Special Cuts
Source:
Asymptotic Differential Algebra and Model Theory of Transseries
Author(s):

Matthias Aschenbrenner

Lou van den Dries

Joris van der Hoeven

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691175423.003.0012

This chapter deals with eventual quantities, immediate extensions, and special cuts. It first considers the behavior of eventual quantities before discussing Newton weight, Newton degree, and Newton multiplicity as well as Newton weight of linear differential operators. It then establishes the following result: Every asymptotically maximal H-asymptotic field with rational asymptotic integration is spherically complete. The chapter proceeds by describing special (definable) cuts in H-asymptotic fields K with asymptotic integration and introducing some key elementary properties of K, namely λ‎-freeness and ω‎-freeness, which indicate that these cuts are not realized in K. It shows that has these properties. Finally, it looks at certain special existentially definable subsets of Liouville closed H-fields K, along with the behavior of the functions ω‎ and λ‎ on these sets.

Keywords:   eventual quantities, immediate extension, special cut, Newton weight, Newton degree, Newton multiplicity, linear differential operator, H-asymptotic field, rational asymptotic integration, Liouville closed H-field

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