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Drinfeld modules #35026

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df3598c
Create a draft of FiniteDrinfeldModule
kryzar Apr 26, 2022
84c7b5b
Create draft for finite Drinfeld module doctest
kryzar Apr 26, 2022
2723f99
Fix FiniteDrinfeldModule action in all.py
kryzar Apr 27, 2022
1e6ec01
Merge branch 'develop' into u/antoine-leudiere/finite_drinfeld_modules
kryzar May 12, 2022
8bef6ba
Refactor _latex_ method in FiniteDrinfeldModule
kryzar May 5, 2022
f0a9f63
Refactor FiniteDrinfeldModuleAction
kryzar May 6, 2022
86348ff
Add various getters for FiniteDrinfeldModule
kryzar May 5, 2022
2c24562
Add methods to deal with complex multiplication in FiniteDrinfeldModule
kryzar May 6, 2022
0499221
Add methods to deal with isogenies in FiniteDrinfeldModule
kryzar May 10, 2022
3609c91
Move FiniteDrinfeldModule to sage.rings.function_field
kryzar May 10, 2022
e3fc37f
Fix method is_morphism in FiniteDrinfeldModule
kryzar May 12, 2022
0515d9b
Add blank line before FiniteDrinfeldModuleAction (PEP8)
kryzar May 12, 2022
8c848d9
Refactor doctest in FiniteDrinfeldModule
kryzar May 11, 2022
7c96d1f
Alphabetically order methods in sage.rings.function_field.finite_drin…
kryzar May 12, 2022
1a7891f
Create class FiniteDrinfeldModule_rank_two
kryzar May 13, 2022
8db54f2
Merge branch 'develop' into u/antoine-leudiere/finite_drinfeld_modules
kryzar May 16, 2022
28d01ed
Merge branch 'develop' into u/antoine-leudiere/finite_drinfeld_modules
kryzar Jun 1, 2022
9bcadf2
Fix j-invariant method for FiniteDrinfeldModule_rank_two
kryzar Jun 9, 2022
3435064
Delete class FiniteDrinfeldModule_rank_two
kryzar Jun 16, 2022
571b80c
Merge branch 'develop' into u/antoine-leudiere/finite_drinfeld_module…
kryzar Jun 16, 2022
a32d290
Make FiniteDrinfeldModule inherit CategoryObject
kryzar Jun 16, 2022
1355069
Fix FiniteDrinfeldModule characteristic polynomial
kryzar Jun 16, 2022
927c47c
Create category DrinfeldModules
kryzar Jun 16, 2022
2537cb3
Define category DrinfeldModules with gamma
kryzar Jun 16, 2022
445efbe
Change DrinfeldModules _repr_
kryzar Jun 16, 2022
d032810
BACKUP COMMIT
kryzar Jun 16, 2022
aa0a384
Merge branch 'reduced_charpoly' into drinfeld_modules
xcaruso Jun 21, 2022
94bad90
characteristic polynomial of Frobenius in all rank
xcaruso Jun 21, 2022
3b47737
Change FiniteDrinfeldModule to DrinfeldModule
kryzar Jun 22, 2022
45be701
Clean DrinfeldModules category
kryzar Jun 22, 2022
0ccdf26
Clean DrinfeldModule constructor
kryzar Jun 22, 2022
92b9e68
Change polring to functions_ring in DrinfeldModule
kryzar Jun 22, 2022
2b01218
Create morphisms for DrinfeldModule
kryzar Jun 22, 2022
b823a57
Remove is_morphism, etc, methods of DrinfeldModule
kryzar Jun 22, 2022
61e9312
Merge branch 'u/antoine-leudiere/finite_drinfeld_modules-category_obj…
xcaruso Jun 23, 2022
c018452
refactor code + classcall_private
xcaruso Jun 23, 2022
b26c813
with all files
xcaruso Jun 23, 2022
22e4f83
Rename morphism.py to homset.py
kryzar Jun 23, 2022
a9848ae
Fix DrinfeldModuleHomset import in DrinfeldModule
kryzar Jun 23, 2022
1bbad0c
Create DrinfeldModuleMorphism
kryzar Jun 29, 2022
009ea5a
Merge branch 'develop' into public/drinfeld_module
kryzar Aug 3, 2022
578ccaa
Rename method functions_ring to function_ring in DrinfeldModule
kryzar Aug 4, 2022
b2e5978
Fix frobenius_charpoly method in FiniteDrinfeldModule
kryzar Aug 4, 2022
72f1642
Remove __eq__ method in DrinfeldModule
kryzar Aug 4, 2022
8f9c18a
Refactor characteristic method in DrinfeldModules
kryzar Aug 4, 2022
1c81b5b
Remove a constructor in DrinfeldModule
kryzar Aug 4, 2022
3fe93da
Change _L to _base_ring in DrinfeldModule
kryzar Aug 4, 2022
01a42cc
Fix velu method in DrinfeldModule
kryzar Aug 4, 2022
c8049b0
Remove duplicate code in DrinfeldModuleMorphism
kryzar Aug 4, 2022
aeeea81
Create frobenius_endomorphism method in FiniteDrinfeldModule
kryzar Aug 4, 2022
055e46c
Comment out frobenius method in DrinfeldModule
kryzar Aug 4, 2022
f8fdad5
Write first draft of DrinfeldModule docstring
kryzar Aug 4, 2022
a1b47ce
Add Drinfeld modules to the reference
kryzar Aug 4, 2022
cac6471
Fix constructor of DrinfeldModuleMorphism
kryzar Aug 5, 2022
d4eddc8
_Minor change
kryzar Aug 5, 2022
4d9e5a0
Create __eq__ method in DrinfeldModuleMorphism
kryzar Aug 5, 2022
fbddd9c
Create is_finite method in DrinfeldModule
kryzar Aug 5, 2022
9ef98d3
Create is_isomorphism method in DrinfeldModuleMorphism
kryzar Aug 8, 2022
f4bb02d
Refactor _repr_ method in DrinfeldModuleHomset
kryzar Aug 5, 2022
a21e526
Remove _check_base_fields function from drinfeld_module.py
kryzar Aug 5, 2022
a7c48a4
Fix __eq__ method in DrinfeldModuleMorphism
kryzar Aug 5, 2022
a77f850
Refactor exception messages in DrinfeldModule
kryzar Aug 7, 2022
a60bede
Rewrite DrinfeldModule docstring
kryzar Aug 5, 2022
a293345
Add classic Drinfeld module references to bib. master file
kryzar Aug 7, 2022
64be056
Merge branch 'develop' into public/drinfeld_module
kryzar Aug 8, 2022
50b0501
Check input morphism in DrinfeldModules constructor
kryzar Aug 10, 2022
73aa69b
Check input in DrinfeldModule constructor
kryzar Aug 10, 2022
f8b7030
Add a note in DrinfeldModule docstring
kryzar Aug 10, 2022
db42b99
Change _domain and _codomain names in DrinfeldModules
kryzar Aug 10, 2022
96776b7
Fix change_ring method in DrinfeldModule class
kryzar Aug 10, 2022
5a64083
Add polcast flag to build Drinfeld modules
kryzar Aug 12, 2022
750d797
(minor) Fix comment in DrinfeldModule constructor
kryzar Aug 13, 2022
ab02ba1
Add name flag to change_ring method in DrinfeldModule
kryzar Aug 13, 2022
4c9b81d
Add examples in DrinfeldModule doc and rewrite docstrings
kryzar Aug 12, 2022
ec1d6bd
Refactor invert method in DrinfeldModule
kryzar Aug 13, 2022
dca2f79
Create method random_element in DrinfeldModules
kryzar Aug 16, 2022
ad98d81
Create method constant_coefficient in DrinfeldModules
kryzar Aug 16, 2022
20e2ead
Refactor _repr_ method in DrinfeldModuleHomset
kryzar Aug 16, 2022
d61a0cf
Apply David's suggestions (ticket 33713 comment 80)
kryzar Aug 16, 2022
f8e34f2
Make DrinfeldModuleMorphism inherit UniqueRepresentation
kryzar Aug 17, 2022
e9b8913
Create _latex_ method in DrinfeldModuleMorphism
kryzar Aug 17, 2022
6e4595b
(fix) Remove delta() in FiniteDrinfeldModule
kryzar Aug 17, 2022
6e5cd2d
Write docstrings for DrinfeldModuleMorphism
kryzar Aug 17, 2022
5decb91
Define __init__ and self._frobenius_trace/norm for FiniteDrinfeldModule
kryzar Aug 18, 2022
daf9b8a
(minor) Add small enhancements to DrinfeldModule doc
kryzar Aug 18, 2022
f16c65a
Write docstrings for FiniteDrinfeldModule
kryzar Aug 18, 2022
a1e569d
Refactor exception messages for Drinfeld modules
kryzar Aug 18, 2022
5e6f4c7
Refactor invert method in DrinfeldModule
kryzar Aug 18, 2022
880ab67
Proofread docstrings in DrinfeldModule
kryzar Aug 18, 2022
999a0ae
Fix function_fields imports
kryzar Aug 18, 2022
942a9c8
Fix DrinfeldModuleAction
kryzar Aug 18, 2022
721f307
Use super().__init__(...) for Drinfeld modules (esp. DrinfeldModuleMo…
kryzar Aug 19, 2022
919cf5e
Create _latex_ method in DrinfeldModuleHomset
kryzar Aug 19, 2022
70c215a
Write docstrings for DrinfeldModuleHomset
kryzar Aug 19, 2022
c9bfc05
(minor) Delete a blank line in DrinfeldModules
kryzar Aug 19, 2022
394d0f5
(minor) Change a sentence in DrinfeldModuleMorphism
kryzar Aug 19, 2022
0f869f2
(minor) Change first sentences of Drinfeld module classes
kryzar Aug 19, 2022
14f98aa
Change _get_action_ to action in DrinfeldModule
kryzar Aug 19, 2022
b014a19
Reorder methods in DrinfeldModule
kryzar Aug 19, 2022
138902d
Write docstrings for DrinfeldModuleAction
kryzar Aug 22, 2022
de5e038
Rename random_element method to random_object in DrinfeldModules
kryzar Aug 22, 2022
e50a5ca
Handle some zero Fq[X]-characteristic special cases in DrinfeldModules
kryzar Aug 22, 2022
f5aa489
Create _base attribute in DrinfeldModules
kryzar Aug 22, 2022
386efec
Write docstrings in DrinfeldModules
kryzar Aug 22, 2022
e80ed4b
Make OUTPUT: fields oneliners in Drinfeld modules classes
kryzar Aug 22, 2022
3962e10
Raise an exception if invalid input for velu method in DrinfeldModule
kryzar Aug 22, 2022
5f8c005
Add _latex_ and _repr_ docstrings in Drinfeld modules classes
kryzar Aug 22, 2022
83c728e
Rename j method to j_invariant in DrinfeldModule
kryzar Aug 22, 2022
53088a9
Add various docstrings in Drinfeld module classes
kryzar Aug 22, 2022
261ab62
Include final comments (87-90) from Ticket
kryzar Aug 22, 2022
c6aaebd
Add coefficients methods in DrinfeldModule
kryzar Aug 22, 2022
529f5ec
Use Fq[X] notation as little as possible in Drinfeld modules classes …
kryzar Aug 23, 2022
8b890f4
Change an exception text in DrinfeldModules
kryzar Aug 23, 2022
3f542d8
Check morphism is non zero in DrinfeldModules
kryzar Aug 23, 2022
fd9126c
Add various TESTS: in Drinfeld modules classes
kryzar Aug 23, 2022
a9152b7
Change validity test in DrinfeldModules method __call__
kryzar Aug 29, 2022
52cb567
Remove method __getitem__ in DrinfeldModule
kryzar Aug 29, 2022
0f3d19c
Proofread docs in Drinfeld module classes
kryzar Aug 29, 2022
ea4a843
Lint Drinfeld modules code
kryzar Aug 29, 2022
5d21c54
Change sage.categories.drinfeld_modules title
kryzar Aug 29, 2022
43c430f
Refactor Drinfeld modules in reference
kryzar Aug 29, 2022
cf503d8
Change _call_ to object in DrinfeldModules
kryzar Sep 8, 2022
f98c968
Change base for Drinfeld modules classes
kryzar Sep 8, 2022
cfad960
Use Parent instead of CategoryObject in DrinfeldModule
kryzar Sep 8, 2022
dbcbaf7
Implement base_ring in DrinfeldModule
kryzar Sep 8, 2022
771f406
Use Category_over_base instead of CategoryWithParameters for Drinfeld…
kryzar Sep 8, 2022
674faeb
Make DrinfeldModuleMorphism inherit Morphism
kryzar Sep 8, 2022
71a6b4a
Merge branch 'develop' into public/drinfeld_module
kryzar Sep 8, 2022
d597953
Remove methods ore_variable from Drinfeld module classes
kryzar Sep 9, 2022
2ca2183
Remove `class ElementMethods` from DrinfeldModules
kryzar Sep 9, 2022
15efff9
(Fix) Add doc of morphism method in DrinfeldModule
kryzar Sep 9, 2022
2e77452
Enhance base and base ring distinctions in doc
kryzar Sep 9, 2022
4412ad1
Move 'category methods' from DrinfeldModule to DrinfeldModules
kryzar Sep 9, 2022
52e6d06
(minor) Change a comment in a doctstring
kryzar Sep 20, 2022
d617406
Add a better error message in DrinfeldModules when the characteristic…
kryzar Sep 20, 2022
1eab9ee
Fix height method
kryzar Sep 20, 2022
bcdfb60
Fix definition of finite drinfeld module in FiniteDrinfeldModule
kryzar Sep 20, 2022
b1ce355
Merge branch 'develop' into public/drinfeld_module
kryzar Sep 20, 2022
0cdd80f
(fix) Fix function_field/all.py
kryzar Sep 21, 2022
8052550
Remove .all import
kryzar Sep 22, 2022
a0285e1
Merge branch 'develop' into public/drinfeld_module
kryzar Sep 27, 2022
1e15803
(minor) Fix typo in docstring
kryzar Sep 27, 2022
5fe2745
Fix lint error E265
kryzar Sep 29, 2022
1e0cfc8
(minor) Remove a blank line in DrinfeldModules
kryzar Oct 3, 2022
e0b22d4
(minor) Change Sage to SageMath in Drinfeld modules reference
kryzar Oct 4, 2022
e9c99e8
(minor) Remove unused import
kryzar Oct 4, 2022
a687972
(minor) Remove useless comment
kryzar Oct 5, 2022
15df110
(fix) Address first part of ticket 33713 comment 123
kryzar Oct 3, 2022
f55ebfe
Emphasis on K infinite in DrinfeldModule docstring
kryzar Oct 3, 2022
a207fdd
Add 100% doctest coverage
kryzar Oct 3, 2022
e042ae2
State that the code is for Drinfeld Fq[X]-modules in DrinfeldModule d…
kryzar Oct 4, 2022
89fa351
(fix) Change a NotImplementedError to ValueError in DrinfeldModuleHomset
kryzar Oct 4, 2022
e0846a3
(fix) In INPUT docstring fields, change `default` to `optional: ...`
kryzar Oct 4, 2022
f71aff4
(fix) Move base_ring from DrinfeldModules.ParentMethods to DrinfeldMo…
kryzar Oct 4, 2022
bf5a4a4
(fix) Replace TESTS: by TESTS:: and EXAMPLES: by EXAMPLES::
kryzar Oct 4, 2022
c821b20
(fix) Fix various small issues in doc and `make doc` (see description)
kryzar Oct 4, 2022
c269135
(fix) Fix DrinfeldModule.base_ring docstring
kryzar Oct 4, 2022
b59d836
(fix) Fix DrinfeldModules.Endsets
kryzar Oct 5, 2022
b763827
Split some code blocks using ::
kryzar Oct 4, 2022
3e4d007
Merge branch 'develop' into public/drinfeld_module
kryzar Oct 5, 2022
7d8e358
(fix) Fix lint check Github job
kryzar Oct 5, 2022
26fe426
(fix) Adress typos from comment 264
kryzar Oct 10, 2022
bee20a0
Add details on the base definition in DrinfeldModule doc
kryzar Oct 11, 2022
78c99e0
Enhance "base ring comment"
kryzar Oct 11, 2022
31e8cf5
Merge branch 'develop' into public/drinfeld_module
kryzar Nov 15, 2022
6982ae0
Merge branch 'public/ring-extension-characteristic' into public/drinf…
kryzar Nov 15, 2022
0374b6d
(breaking) Change base morphism to base ring for Drinfeld modules
kryzar Nov 15, 2022
ca22646
Merge branch 'develop' into drinfeld-module
kryzar Feb 6, 2023
1ef1412
Remove `# todo: not tested` comments
kryzar Feb 6, 2023
050c2f6
(minor) Typo
kryzar Feb 6, 2023
e24c366
Change X to T
kryzar Feb 6, 2023
774f670
Adress David's comments
kryzar Feb 6, 2023
a70c248
Add a LaTeX name for Drinfeld modules
kryzar Feb 6, 2023
b48243c
(minor) Modify DrinfeldModule.invert docstring
kryzar Feb 6, 2023
b8e5495
(minor) Typo
kryzar Feb 6, 2023
ef94f98
Proofread DrinfeldModuleAction
kryzar Feb 6, 2023
9c34fe1
Major typo fix
ymusleh Feb 6, 2023
6601038
More extremely critical changes
ymusleh Feb 6, 2023
b4371c8
Manage merge conflict
kryzar Feb 6, 2023
5e3182e
Adress David's suggestions for DrinfeldModuleHomset
kryzar Feb 6, 2023
09f27ab
Huge fix people from paris wouldn't understand
ymusleh Feb 6, 2023
4732e5d
9.8.rc0 -> 9.8.rc1
xcaruso Feb 7, 2023
1863590
fix creation of the ring extension when the top ring is itself an ext…
xcaruso Feb 7, 2023
7cc1eb6
Refactor DrinfeldModule docstring (see details)
kryzar Feb 7, 2023
63fc7d9
Various changes to doc (see details)
kryzar Feb 7, 2023
53fab7d
Handle NotImplementedError for is_subring
kryzar Feb 7, 2023
09b6328
Adress David's suggestions for Morphism
kryzar Feb 7, 2023
fc9b281
Try to cast the base morphism to a natural map, when possible
kryzar Feb 7, 2023
bd430de
Fix LaTeX name init
kryzar Feb 7, 2023
d8441f5
(fix) Stop using base.codomain
kryzar Feb 7, 2023
cd96c6f
Fix some doctests
kryzar Feb 7, 2023
ba4fc76
Modify natural map creation
kryzar Feb 7, 2023
8c55708
Merge branch 'drinfeld-module' of github:kryzar/sage into drinfeld-mo…
xcaruso Feb 7, 2023
7bc3800
Fix extensions over Fq
kryzar Feb 7, 2023
2407ba0
Create base_over_constants_field DrinfeldModules method
kryzar Feb 7, 2023
88505c2
Create base_over_constants_field DrinfeldModule (singular) method
kryzar Feb 7, 2023
3cea902
(fix) Remove `not tested` comments from DrinfeldModules
kryzar Feb 7, 2023
eb42cbc
Fix invert method and move it for *finite* Drinfeld modules
kryzar Feb 7, 2023
de74a78
(fix) Identation problem in DrinfeldModules
kryzar Feb 7, 2023
506fb15
(fix) Fix some doctest fails
kryzar Feb 7, 2023
301aa0a
(fix) Fix DrinfeldModules failing doctests
kryzar Feb 7, 2023
90aa9ac
Enhance invert method code
kryzar Feb 7, 2023
d954bc3
Lighten DrinfeldModule representation
kryzar Feb 7, 2023
3b2314b
small changes
xcaruso Feb 7, 2023
4ccc577
category
xcaruso Feb 8, 2023
b7f0ecf
Fix stuff
kryzar Feb 8, 2023
30617f9
Add hash methods
kryzar Feb 8, 2023
3902255
(fix) Typo
kryzar Feb 8, 2023
4afccce
Merge branch 'develop' into drinfeld-module
kryzar Feb 8, 2023
fbaf715
Refactor latexname
kryzar Feb 9, 2023
1487984
Enhance an error message
kryzar Feb 9, 2023
0e6cee3
(fix) Fix PEP8 typo
kryzar Feb 9, 2023
53cbf19
(fix) Fix an indentation problem in the doc
kryzar Feb 15, 2023
4c5666a
Merge branch 'develop' into drinfeld-module
kryzar Feb 15, 2023
24ad78a
Merge branch 'develop' into drinfeld-module
xcaruso Feb 20, 2023
93b5cf4
Merge branch 'drinfeld-module' of github:kryzar/sage into drinfeld-mo…
xcaruso Feb 20, 2023
11a4fa9
Merge branch 'develop' into drinfeld-module
kryzar Feb 27, 2023
fbf1e09
Fix master bib. file entries
kryzar Feb 28, 2023
1346b21
Fix doc typos
kryzar Feb 28, 2023
cda700f
Merge branch 'develop' into drinfeld-module
kryzar Mar 3, 2023
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1 change: 1 addition & 0 deletions src/doc/en/reference/drinfeld_modules/conf.py
42 changes: 42 additions & 0 deletions src/doc/en/reference/drinfeld_modules/index.rst
@@ -0,0 +1,42 @@
Drinfeld modules
====================================

SageMath include facilities to manipulate Drinfeld modules and their morphisms. The
main entry point is the class
:class:`sage.rings.function_field.drinfeld_modules.drinfeld_module.DrinfeldModule`.

Drinfeld modules
----------------

.. toctree::
:maxdepth: 2

sage/rings/function_field/drinfeld_modules/drinfeld_module
sage/rings/function_field/drinfeld_modules/finite_drinfeld_module

Morphisms and isogenies
-----------------------

.. toctree::
:maxdepth: 2

sage/rings/function_field/drinfeld_modules/morphism
sage/rings/function_field/drinfeld_modules/homset

The module action induced by a Drinfeld module
----------------------------------------------

.. toctree::
:maxdepth: 2

sage/rings/function_field/drinfeld_modules/action

The category of Drinfeld modules
--------------------------------

.. toctree::
:maxdepth: 2

sage/categories/drinfeld_modules

.. include:: ../footer.txt
1 change: 1 addition & 0 deletions src/doc/en/reference/index.rst
Expand Up @@ -115,6 +115,7 @@ Number Fields, Function Fields, and Valuations
* :doc:`Number Fields <number_fields/index>`
* :doc:`Function Fields <function_fields/index>`
* :doc:`Discrete Valuations <valuations/index>`
* :doc:`Drinfeld Modules <drinfeld_modules/index>`

Number Theory
-------------
Expand Down
16 changes: 16 additions & 0 deletions src/doc/en/reference/references/index.rst
Expand Up @@ -2550,6 +2550,9 @@ REFERENCES:
TR-737-05, (2005).
ftp://ftp.cs.princeton.edu/reports/2005/737.pdf

.. [Gek1991] \E.-U. Gekeler. On finite Drinfeld modules. Journal of
algebra, 1(141):187–203, 1991.

.. [GG2012] Jim Geelen and Bert Gerards, Characterizing graphic
matroids by a system of linear equations,
submitted, 2012. Preprint:
Expand Down Expand Up @@ -2713,6 +2716,9 @@ REFERENCES:
.. [Gos1972] Bill Gosper, "Continued Fraction Arithmetic"
https://perl.plover.com/classes/cftalk/INFO/gosper.txt

.. [Gos1998] \D. Goss. Basic structures of function field arithmetic. Springer,
1998.

.. [Gor1980] Daniel Gorenstein, Finite Groups (New York: Chelsea
Publishing, 1980)

Expand Down Expand Up @@ -4569,6 +4575,11 @@ REFERENCES:
Int. Math. Res. Not. (2015).
:doi:`10.1093/imrn/rnv194`, :arxiv:`1408.0320`.

.. [MS2019] \Y. Musleh and É. Schost. Computing the characteristic polynomial
of a finite rank two Drinfeld module. In Proceedings of the 2019
ACM on International Symposium on Symbolic and Algebraic
Computation, pages 307–314. ACM, 2019.

.. [MSSY2001] Mateescu, A., Salomaa, A., Salomaa, K. and Yu, S., *A
sharpening of the Parikh mapping*. Theoret. Informatics Appl. 35
(2001) 551-564.
Expand Down Expand Up @@ -5149,6 +5160,8 @@ REFERENCES:
.. [Ros2002] Rosenfeld, Vladimir Raphael, 2002: Enumerating De Bruijn
Sequences. *Communications in Math. and in Computer Chem.*

.. [Rosen2002] \M. Rosen. Number theory in function fields. Springer, 2022.

.. [Rot2001] Gunter Rote, *Division-Free Algorithms for the
Determinant and the Pfaffian: Algebraic and Combinatorial
Approaches*, H. Alt (Ed.): Computational Discrete
Expand Down Expand Up @@ -6011,6 +6024,9 @@ REFERENCES:
.. [Voi2012] \J. Voight. Identifying the matrix ring: algorithms for
quaternion algebras and quadratic forms, to appear.

.. [VS06] \G.D. Villa Salvador. Topics in the Theory of Algebraic Function
Fields. Birkhäuser, 2006.

.. [VW1994] Leonard Van Wyk. *Graph groups are biautomatic*. J. Pure
Appl. Alg. **94** (1994). no. 3, 341-352.

Expand Down