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[Merged by Bors] - doc(ring_theory/hahn_series): Update Hahn Series docstring #8883

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[Merged by Bors] - doc(ring_theory/hahn_series): Update Hahn Series docstring #8883

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11 changes: 11 additions & 0 deletions docs/references.bib
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Expand Up @@ -1155,6 +1155,17 @@ @Article{ Vaisala_2003
doi = {10.2307/3647749}
}

@Article{ van_der_hoeven,
author = {van der Hoeven, Joris},
year = {2001},
month = {12},
pages = {},
title = {Operators on generalized power series},
volume = {45},
journal = {Illinois Journal of Mathematics},
doi = {10.1215/ijm/1258138061}
}

@Book{ wall2018analytic,
title = {Analytic Theory of Continued Fractions},
author = {Wall, H.S.},
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18 changes: 15 additions & 3 deletions src/ring_theory/hahn_series.lean
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Expand Up @@ -11,6 +11,14 @@ import ring_theory.power_series.basic

/-!
# Hahn Series
If `Γ` is ordered and `R` has zero, then `hahn_series Γ R` consists of formal series over `Γ` with
coefficients in `R`, whose supports are partially well-ordered. With further structure on `R` and
`Γ`, we can add further structure on `hahn_series Γ R`, with the most studied case being when `Γ` is
a linearly ordered abelian group and `R` is a field, in which case `hahn_series Γ R` is a
valued field, with value group `Γ`.

These generalize Laurent series (with value group `ℤ`), and Laurent series are implemented that way
in the file `ring_theory/laurent_series`.

## Main Definitions
* If `Γ` is ordered and `R` has zero, then `hahn_series Γ R` consists of
Expand All @@ -26,11 +34,15 @@ import ring_theory.power_series.basic
topology, because there are topologically summable families that do not satisfy the axioms of
`hahn_series.summable_family`, and formally summable families whose sums do not converge
topologically.
* Laurent series over `R` are implemented as `hahn_series ℤ R` in the file
`ring_theory/laurent_series`.

## TODO
* Given `[linear_ordered_add_comm_group Γ]` and `[field R]`, define `field (hahn_series Γ R)`.
* Build an API for the variable `X`
* Define Laurent series
* Build an API for the variable `X` (defined to be `single 1 1 : hahn_series Γ R`) in analogy to
`X : polynomial R` and `X : power_series R`

## References
- [J. van der Hoeven, *Operators on Generalized Power Series*][van_der_hoeven]

-/

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