
Contents
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Pre-d-valued fields by coarsening Pre-d-valued fields by coarsening
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Specialization Specialization
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Extensions of pre-differential-valued fields I Extensions of pre-differential-valued fields I
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Extensions of pre-differential-valued fields II Extensions of pre-differential-valued fields II
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Notes and comments Notes and comments
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10.2 Adjoining Integrals 10.2 Adjoining Integrals
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Notes and comments Notes and comments
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10.3 THE DIFFERENTIAL-VALUED HULL 10.3 THE DIFFERENTIAL-VALUED HULL
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The proof of the theorem above provides extra information: The proof of the theorem above provides extra information:
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Notes and comments Notes and comments
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Notes and comments Notes and comments
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Algebraic extensions of pre-H-fields Algebraic extensions of pre-H-fields
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Immediate extensions of pre-H-fields Immediate extensions of pre-H-fields
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Adjoining integrals Adjoining integrals
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Extending the constant field Extending the constant field
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Adjoining exponential integrals Adjoining exponential integrals
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Notes and comments Notes and comments
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Completion Completion
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The main result about Liouville closures The main result about Liouville closures
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Liouville towers Liouville towers
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Uniqueness of Liouville closure Uniqueness of Liouville closure
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10.7 Miscellaneous Facts about Asymptotic Fields 10.7 Miscellaneous Facts about Asymptotic Fields
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Differentially closed fields cannot be asymptotic Differentially closed fields cannot be asymptotic
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The logarithmic derivative map on an elliptic curve The logarithmic derivative map on an elliptic curve
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An asymptotic field that is not a coarsening of a pre-d-valued field An asymptotic field that is not a coarsening of a pre-d-valued field
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Definability of T in T[i] Definability of T in T[i]
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Cite
Abstract
This chapter considers H-fields, pre-differential-valued fields with a field ordering that interacts with the valuation and derivation. Axiomatizing this interaction yields the notion of a pre-H-field; H-fields are d-valued pre-H-fields. The chapter begins by upgrading some basic facts on asymptotic fields to pre-d-valued fields; for example, algebraic extensions of pre-d-valued fields are pre-d-valued, not just asymptotic. It then adjoins integrals to pre-d-valued fields of H-type. It shows that every pre-d-valued field of H-type has a canonical differential-valued extension. It also adjoins exponential integrals to pre-d-valued fields of H-type. Finally, it describes Liouville closed H-fields, and especially the uniqueness properties of Liouville closure.
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