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Preprint Number 1547
1547. Pantelis E. Eleftheriou, Alex Savatovsky
Expansions of real closed fields which introduce no new smooth functions
Submission date: 25 December 2018
We prove the following theorem: let R' be an expansion of the real field R, such that every definable set (I) is a uniform countable union of semialgebraic sets, and (II) contains a semialgebraic chunk. Then every definable smooth function with open semialgebraic domain is semialgebraic.
Conditions (I) and (II) hold for various d-minimal expansions R'=(R, P) of the real field, such as when P=2^Z, or P is an iteration sequence. A generalization of the theorem to d-minimal expansions R' of R_an fails. On the other hand, we prove our theorem for expansions R' of arbitrary real closed fields. Moreover, its conclusion holds for certain structures with d-minimal open core, such as (R, R_alg, 2^Z).
Mathematics Subject Classification: 03C64
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