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# Byte-compiled / optimized / DLL files
__pycache__/
*.py[cod]
*$py.class
# C extensions
*.so
# Distribution / packaging
.Python
env/
build/
develop-eggs/
dist/
downloads/
eggs/
.eggs/
lib/
lib64/
parts/
sdist/
var/
*.egg-info/
.installed.cfg
*.egg
# PyInstaller
# Usually these files are written by a python script from a template
# before PyInstaller builds the exe, so as to inject date/other infos into it.
*.manifest
*.spec
# Installer logs
pip-log.txt
pip-delete-this-directory.txt
# Unit test / coverage reports
htmlcov/
.tox/
.coverage
.coverage.*
.cache
nosetests.xml
coverage.xml
*,cover
.hypothesis/
# Translations
*.mo
*.pot
# Django stuff:
*.log
local_settings.py
# Flask stuff:
instance/
.webassets-cache
# Scrapy stuff:
.scrapy
# Sphinx documentation
docs/_build/
# PyBuilder
target/
# IPython Notebook
.ipynb_checkpoints
# pyenv
.python-version
# celery beat schedule file
celerybeat-schedule
# dotenv
.env
# virtualenv
venv/
ENV/
# Spyder project settings
.spyderproject
# Rope project settings
.ropeproject
/.travis_ci_gh_pages_deploy_key
/.travis_ci_gh_pages_deploy_key.pub
/gh-pages

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# Dockerfile for binder
# Reference: https://mybinder.readthedocs.io/en/latest/dockerfile.html#preparing-your-dockerfile
FROM sagemath/sagemath:latest
# Copy the contents of the repo in ${HOME}
COPY --chown=sage:sage . ${HOME}
# Install this package and dependencies
RUN sage -pip install .

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GNU GENERAL PUBLIC LICENSE
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state the exclusion of warranty; and each file should have at least
the "copyright" line and a pointer to where the full notice is found.
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Copyright (C) <year> <name of author>
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it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
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but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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along with this program. If not, see <https://www.gnu.org/licenses/>.
Also add information on how to contact you by electronic and paper mail.
If the program does terminal interaction, make it output a short
notice like this when it starts in an interactive mode:
<program> Copyright (C) <year> <name of author>
This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
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under certain conditions; type `show c' for details.
The hypothetical commands `show w' and `show c' should show the appropriate
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64
README.rst Normal file
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==============
Vector Bundles
==============
This is a `SageMath <http://www.sagemath.org>`_ implementing algorithms for creating and
manipulating Vector Bundles over algebraic curves on finite field. All manipulations are
done using algebra on function fields.
Installation
------------
Local install from source
^^^^^^^^^^^^^^^^^^^^^^^^^
Download the source from the git repository::
$ git clone https://github.com/sagemath/sage_sample.git
Change to the root directory and run::
$ sage -pip install --upgrade --no-index -v .
For convenience this package contains a `makefile <makefile>`_ with this
and other often used commands. Should you wish too, you can use the
shorthand::
$ make install
Usage
-----
Once the package is installed, you can use it in Sage with::
sage: from vector_bundle import *
See also the `demo notebook <demo.ipynb>`_ and the documentation.
Documentation
-------------
The documentation of the package can be generated using Sage's
``Sphinx`` installation::
$ cd docs
$ sage -sh -c "make html"
Shorthand::
$ make doc
TEST
----
To run the test suite of the package, simply run the command::
$make test
from the root of the repository.
CONTACT
-------
Mickaël Montessinos: mickael.montessinos@mif.vu.lt

1
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0.1.0

234
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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# A demo of the vector_bundle package."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"from vector_bundle import *\n",
"F.<x> = FunctionField(GF(101))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Constructing an indecomposable bundle on an elliptic curve\n",
"We construct an indecomposable vector bundle of rank 5 and degree 3 on a function field of genus 1. \n",
"This construction may be done automatically using the ```atiyah_bundle``` function but we break it down step by step."
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[Ideal (1) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x]\n",
"[(1)]\n",
"[((x/(x^2 + 1))*y)]\n"
]
}
],
"source": [
"R.<y> = F[]\n",
"K.<y> = F.extension(y^2 - x^3 - x)\n",
"deg_1_bundle = VectorBundle(K, K.places_infinite()[0].divisor())\n",
"E = deg_1_bundle\n",
"print(E.coefficient_ideals())\n",
"print(E.basis_finite())\n",
"print(E.basis_infinite())"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[Ideal (x^2/(x^2 + 3)) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (1) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x]\n",
"[(1, 0), (0, 1)]\n",
"[(1, 0), (100*x^3/(x^2 + 1), (x/(x^2 + 1))*y)]\n"
]
}
],
"source": [
"E = E.extension_by_global_sections()\n",
"print(E.coefficient_ideals())\n",
"print(E.basis_finite())\n",
"print(E.basis_infinite())"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[Ideal (x^2/(x^2 + 3)) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (1) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x]\n",
"[(1, 0), (0, 1)]\n",
"[((x/(x^2 + 1))*y, 0), ((100*x^4/(x^4 + 2*x^2 + 1))*y, x^3/(x^2 + 1))]\n"
]
}
],
"source": [
"E = E.tensor_product(deg_1_bundle)\n",
"print(E.coefficient_ideals())\n",
"print(E.basis_finite())\n",
"print(E.basis_infinite())"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[Ideal (x^2/(x^2 + 3)) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (x^2/(x^2 + 3)) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (x^2/(x^2 + 3)) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (x^2/(x^2 + 3)) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (1) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x]\n",
"[(1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)]\n",
"[(1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (100*x^3/(x^2 + 1), 0, 100/x^2*y, (x/(x^2 + 1))*y, 0), (x^6/(x^4 + 2*x^2 + 1), (100*x^4/(x^4 + 2*x^2 + 1))*y, (x/(x^2 + 1))*y, (100*x^4/(x^4 + 2*x^2 + 1))*y, x^3/(x^2 + 1))]\n"
]
}
],
"source": [
"E = E.extension_by_global_sections()\n",
"print(E.coefficient_ideals())\n",
"print(E.basis_finite())\n",
"print(E.basis_infinite())"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"5\n",
"3\n"
]
}
],
"source": [
"print(E.rank())\n",
"print(E.degree())"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We check the algebra of global endomorphisms of E to ensure that it is indecomposable."
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[\n",
"[1 0 0 0 0]\n",
"[0 1 0 0 0]\n",
"[0 0 1 0 0]\n",
"[0 0 0 1 0]\n",
"[0 0 0 0 1]\n",
"]"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"E.end().h0()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Constructing a weakly stable vector bundle following Savin's method\n",
"We construct a weakly stable vector bundle of rank 3 and degree 5 by successive extensions by line bundles. \n",
"See [Sav07] in the references for details. \n",
"This, again, may be achieved directly using the ```savin_bundle``` function."
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"3\n",
"5\n",
"[Ideal (1, 1/x*y) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (1) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (1) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x]\n",
"[(1, 0, 0), (0, 1, 0), (0, 0, 1)]\n",
"[(1, 0, 0), ((100*x/(x^2 + 1))*y, x^3/(x^2 + 1), 0), (0, (100*x^4/(x^4 + 2*x^2 + 1))*y, x^3/(x^2 + 1))]\n"
]
}
],
"source": [
"F = VectorBundle(K, 2*K.places_infinite()[0].divisor())\n",
"F1 = VectorBundle(K, K.places_finite()[0].divisor())\n",
"E1 = F1\n",
"E2 = F.non_trivial_extension(E1)\n",
"E3 = F.non_trivial_extension(E2)\n",
"print(E3.rank())\n",
"print(E3.degree())\n",
"print(E3.coefficient_ideals())\n",
"print(E3.basis_finite())\n",
"print(E3.basis_infinite())"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "SageMath 10.3.beta8",
"language": "sage",
"name": "sagemath"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.11.2"
}
},
"nbformat": 4,
"nbformat_minor": 4
}

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# Makefile for Sphinx documentation
#
# You can set these variables from the command line.
SPHINXOPTS =
SPHINXBUILD = sphinx-build
PAPER =
BUILDDIR = build
# User-friendly check for sphinx-build
ifeq ($(shell which $(SPHINXBUILD) >/dev/null 2>&1; echo $$?), 1)
$(error The '$(SPHINXBUILD)' command was not found. Make sure you have Sphinx installed, then set the SPHINXBUILD environment variable to point to the full path of the '$(SPHINXBUILD)' executable. Alternatively you can add the directory with the executable to your PATH. If you don't have Sphinx installed, grab it from http://sphinx-doc.org/)
endif
# Internal variables.
PAPEROPT_a4 = -D latex_paper_size=a4
PAPEROPT_letter = -D latex_paper_size=letter
ALLSPHINXOPTS = -d $(BUILDDIR)/doctrees $(PAPEROPT_$(PAPER)) $(SPHINXOPTS) source
# the i18n builder cannot share the environment and doctrees with the others
I18NSPHINXOPTS = $(PAPEROPT_$(PAPER)) $(SPHINXOPTS) source
.PHONY: help clean html dirhtml singlehtml pickle json htmlhelp qthelp devhelp epub latex latexpdf text man changes linkcheck doctest gettext
help:
@echo "Please use \`make <target>' where <target> is one of"
@echo " html to make standalone HTML files"
@echo " dirhtml to make HTML files named index.html in directories"
@echo " singlehtml to make a single large HTML file"
@echo " pickle to make pickle files"
@echo " json to make JSON files"
@echo " htmlhelp to make HTML files and a HTML help project"
@echo " qthelp to make HTML files and a qthelp project"
@echo " devhelp to make HTML files and a Devhelp project"
@echo " epub to make an epub"
@echo " latex to make LaTeX files, you can set PAPER=a4 or PAPER=letter"
@echo " latexpdf to make LaTeX files and run them through pdflatex"
@echo " latexpdfja to make LaTeX files and run them through platex/dvipdfmx"
@echo " text to make text files"
@echo " man to make manual pages"
@echo " texinfo to make Texinfo files"
@echo " info to make Texinfo files and run them through makeinfo"
@echo " gettext to make PO message catalogs"
@echo " changes to make an overview of all changed/added/deprecated items"
@echo " xml to make Docutils-native XML files"
@echo " pseudoxml to make pseudoxml-XML files for display purposes"
@echo " linkcheck to check all external links for integrity"
@echo " doctest to run all doctests embedded in the documentation (if enabled)"
clean:
rm -rf $(BUILDDIR)/*
html:
$(SPHINXBUILD) -b html $(ALLSPHINXOPTS) $(BUILDDIR)/html
@echo
@echo "Build finished. The HTML pages are in $(BUILDDIR)/html."
dirhtml:
$(SPHINXBUILD) -b dirhtml $(ALLSPHINXOPTS) $(BUILDDIR)/dirhtml
@echo
@echo "Build finished. The HTML pages are in $(BUILDDIR)/dirhtml."
singlehtml:
$(SPHINXBUILD) -b singlehtml $(ALLSPHINXOPTS) $(BUILDDIR)/singlehtml
@echo
@echo "Build finished. The HTML page is in $(BUILDDIR)/singlehtml."
pickle:
$(SPHINXBUILD) -b pickle $(ALLSPHINXOPTS) $(BUILDDIR)/pickle
@echo
@echo "Build finished; now you can process the pickle files."
json:
$(SPHINXBUILD) -b json $(ALLSPHINXOPTS) $(BUILDDIR)/json
@echo
@echo "Build finished; now you can process the JSON files."
htmlhelp:
$(SPHINXBUILD) -b htmlhelp $(ALLSPHINXOPTS) $(BUILDDIR)/htmlhelp
@echo
@echo "Build finished; now you can run HTML Help Workshop with the" \
".hhp project file in $(BUILDDIR)/htmlhelp."
qthelp:
$(SPHINXBUILD) -b qthelp $(ALLSPHINXOPTS) $(BUILDDIR)/qthelp
@echo
@echo "Build finished; now you can run "qcollectiongenerator" with the" \
".qhcp project file in $(BUILDDIR)/qthelp, like this:"
@echo "# qcollectiongenerator $(BUILDDIR)/qthelp/slabbe.qhcp"
@echo "To view the help file:"
@echo "# assistant -collectionFile $(BUILDDIR)/qthelp/slabbe.qhc"
devhelp:
$(SPHINXBUILD) -b devhelp $(ALLSPHINXOPTS) $(BUILDDIR)/devhelp
@echo
@echo "Build finished."
@echo "To view the help file:"
@echo "# mkdir -p $$HOME/.local/share/devhelp/slabbe"
@echo "# ln -s $(BUILDDIR)/devhelp $$HOME/.local/share/devhelp/slabbe"
@echo "# devhelp"
epub:
$(SPHINXBUILD) -b epub $(ALLSPHINXOPTS) $(BUILDDIR)/epub
@echo
@echo "Build finished. The epub file is in $(BUILDDIR)/epub."
latex:
$(SPHINXBUILD) -b latex $(ALLSPHINXOPTS) $(BUILDDIR)/latex
@echo
@echo "Build finished; the LaTeX files are in $(BUILDDIR)/latex."
@echo "Run \`make' in that directory to run these through (pdf)latex" \
"(use \`make latexpdf' here to do that automatically)."
latexpdf:
$(SPHINXBUILD) -b latex $(ALLSPHINXOPTS) $(BUILDDIR)/latex
@echo "Running LaTeX files through pdflatex..."
$(MAKE) -C $(BUILDDIR)/latex all-pdf
@echo "pdflatex finished; the PDF files are in $(BUILDDIR)/latex."
latexpdfja:
$(SPHINXBUILD) -b latex $(ALLSPHINXOPTS) $(BUILDDIR)/latex
@echo "Running LaTeX files through platex and dvipdfmx..."
$(MAKE) -C $(BUILDDIR)/latex all-pdf-ja
@echo "pdflatex finished; the PDF files are in $(BUILDDIR)/latex."
text:
$(SPHINXBUILD) -b text $(ALLSPHINXOPTS) $(BUILDDIR)/text
@echo
@echo "Build finished. The text files are in $(BUILDDIR)/text."
man:
$(SPHINXBUILD) -b man $(ALLSPHINXOPTS) $(BUILDDIR)/man
@echo
@echo "Build finished. The manual pages are in $(BUILDDIR)/man."
texinfo:
$(SPHINXBUILD) -b texinfo $(ALLSPHINXOPTS) $(BUILDDIR)/texinfo
@echo
@echo "Build finished. The Texinfo files are in $(BUILDDIR)/texinfo."
@echo "Run \`make' in that directory to run these through makeinfo" \
"(use \`make info' here to do that automatically)."
info:
$(SPHINXBUILD) -b texinfo $(ALLSPHINXOPTS) $(BUILDDIR)/texinfo
@echo "Running Texinfo files through makeinfo..."
make -C $(BUILDDIR)/texinfo info
@echo "makeinfo finished; the Info files are in $(BUILDDIR)/texinfo."
gettext:
$(SPHINXBUILD) -b gettext $(I18NSPHINXOPTS) $(BUILDDIR)/locale
@echo
@echo "Build finished. The message catalogs are in $(BUILDDIR)/locale."
changes:
$(SPHINXBUILD) -b changes $(ALLSPHINXOPTS) $(BUILDDIR)/changes
@echo
@echo "The overview file is in $(BUILDDIR)/changes."
linkcheck:
$(SPHINXBUILD) -b linkcheck $(ALLSPHINXOPTS) $(BUILDDIR)/linkcheck
@echo
@echo "Link check complete; look for any errors in the above output " \
"or in $(BUILDDIR)/linkcheck/output.txt."
doctest:
$(SPHINXBUILD) -b doctest $(ALLSPHINXOPTS) $(BUILDDIR)/doctest
@echo "Testing of doctests in the sources finished, look at the " \
"results in $(BUILDDIR)/doctest/output.txt."
xml:
$(SPHINXBUILD) -b xml $(ALLSPHINXOPTS) $(BUILDDIR)/xml
@echo
@echo "Build finished. The XML files are in $(BUILDDIR)/xml."
pseudoxml:
$(SPHINXBUILD) -b pseudoxml $(ALLSPHINXOPTS) $(BUILDDIR)/pseudoxml
@echo
@echo "Build finished. The pseudo-XML files are in $(BUILDDIR)/pseudoxml."

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# -*- coding: utf-8 -*-
#
# sample documentation build configuration file,
# inspried by slabbe configuration file created sphinx-quickstart
#
# This file is execfile()d with the current directory set to its
# containing dir.
#
# Note that not all possible configuration values are present in this
# autogenerated file.
#
# All configuration values have a default; values that are commented out
# serve to show the default.
# General information about the project.
project = u"A SageMath package implementing vector bundles on algebraic curves using only function fields"
copyright = u'2024, Mickaël Montessinos'
package_name = 'vector_bundle'
package_folder = "../../"
authors = u"Mickaël Montessinos"
import six
import sys
import os
from sage.env import SAGE_DOC_SRC, SAGE_DOC, SAGE_SRC
try:
import sage.all
except ImportError:
raise RuntimeError("to build the documentation you need to be inside a Sage shell (run first the command 'sage -sh' in a shell")
# If extensions (or modules to document with autodoc) are in another directory,
# add these directories to sys.path here. If the directory is relative to the
# documentation root, use os.path.abspath to make it absolute, like shown here.
sys.path.insert(0, os.path.abspath(package_folder))
#sys.path.append(os.path.join(SAGE_SRC, "sage_setup", "docbuild", "ext"))
print("Using sys.path = {}".format(sys.path))
# -- General configuration ------------------------------------------------
# If your documentation needs a minimal Sphinx version, state it here.
#needs_sphinx = '1.0'
# Add any Sphinx extension module names here, as strings. They can be
# extensions coming with Sphinx (named 'sphinx.ext.*') or your custom
# ones.
extensions = [
'sphinx.ext.autodoc',
#'sage_autodoc', ## Not available on conda-forge sage!
#'sage_package.sphinx',
'sphinx.ext.doctest',
'sphinx.ext.coverage',
'sphinx.ext.extlinks',
'matplotlib.sphinxext.plot_directive',
#'sphinxcontrib.bibtex'
]
### from Sage src/doc/common/conf.py
# This code is executed before each ".. PLOT::" directive in the Sphinx
# documentation. It defines a 'sphinx_plot' function that displays a Sage object
# through matplotlib, so that it will be displayed in the HTML doc.
plot_html_show_source_link = False
plot_pre_code = """
def sphinx_plot(graphics, **kwds):
import matplotlib.image as mpimg
from sage.misc.temporary_file import tmp_filename
import matplotlib.pyplot as plt
## Option handling is taken from Graphics.save
try:
from sage.plot.multigraphics import GraphicsArray
except ImportError:
from sage.plot.graphics import GraphicsArray
options = dict()
if not isinstance(graphics, GraphicsArray):
options.update(graphics.SHOW_OPTIONS)
options.update(graphics._extra_kwds)
options.update(kwds)
dpi = options.pop('dpi', None)
transparent = options.pop('transparent', None)
fig_tight = options.pop('fig_tight', None)
figsize = options.pop('figsize', None)
## figsize handling is taken from Graphics.matplotlib()
if figsize is not None and not isinstance(figsize, (list, tuple)):
# in this case, figsize is a number and should be positive
try:
figsize = float(figsize) # to pass to mpl
except TypeError:
raise TypeError("figsize should be a positive number, not {0}".format(figsize))
if figsize > 0:
default_width, default_height=rcParams['figure.figsize']
figsize=(figsize, default_height*figsize/default_width)
else:
raise ValueError("figsize should be positive, not {0}".format(figsize))
if figsize is not None:
# then the figsize should be two positive numbers
if len(figsize) != 2:
raise ValueError("figsize should be a positive number "
"or a list of two positive numbers, not {0}".format(figsize))
figsize = (float(figsize[0]),float(figsize[1])) # floats for mpl
if not (figsize[0] > 0 and figsize[1] > 0):
raise ValueError("figsize should be positive numbers, "
"not {0} and {1}".format(figsize[0],figsize[1]))
plt.figure(figsize=figsize)
if isinstance(graphics, GraphicsArray):
## from GraphicsArray.save
figure = plt.gcf()
rows = graphics.nrows()
cols = graphics.ncols()
for i, g in enumerate(graphics):
subplot = figure.add_subplot(rows, cols, i + 1)
g_options = copy(options)
g_options.update(g.SHOW_OPTIONS)
g_options.update(g._extra_kwds)
g_options.pop('dpi', None)
g_options.pop('transparent', None)
g_options.pop('fig_tight', None)
g.matplotlib(figure=figure, sub=subplot, **g_options)
else:
figure = graphics.matplotlib(figure=plt.gcf(), figsize=figsize, **options)
plt.tight_layout(pad=0)
plt.margins(0)
plt.show()
from sage.all_cmdline import *
"""
plot_html_show_formats = False
plot_formats = ['svg', 'pdf', 'png']
# Add any paths that contain templates here, relative to this directory.
# templates_path = ['_templates']
templates_path = [os.path.join(SAGE_DOC_SRC, 'common', 'templates'), '_templates']
# The suffix of source filenames.
source_suffix = '.rst'
# The encoding of source files.
#source_encoding = 'utf-8-sig'
# The master toctree document.
master_doc = 'index'
# The version info for the project you're documenting, acts as replacement for
# |version| and |release|, also used in various other places throughout the
# built documents.
#
from pkg_resources import get_distribution, DistributionNotFound
# The full version, including alpha/beta/rc tags.
try:
release = get_distribution('sage-numerical-interactive-mip').version
except DistributionNotFound:
release = "0.2"
#print("############# release reported: {} ##################".format(release))
# The short X.Y version.
version = '.'.join(release.split('.')[:2])
# The language for content autogenerated by Sphinx. Refer to documentation
# for a list of supported languages.
#language = None
# There are two options for replacing |today|: either, you set today to some
# non-false value, then it is used:
#today = ''
# Else, today_fmt is used as the format for a strftime call.
#today_fmt = '%B %d, %Y'
# List of patterns, relative to source directory, that match files and
# directories to ignore when looking for source files.
exclude_patterns = []
# The reST default role (used for this markup: `text`) to use for all
# documents.
default_role = 'math'
# If true, '()' will be appended to :func: etc. cross-reference text.
#add_function_parentheses = True
# If true, the current module name will be prepended to all description
# unit titles (such as .. function::).
#add_module_names = True
# If true, sectionauthor and moduleauthor directives will be shown in the
# output. They are ignored by default.
#show_authors = False
# The name of the Pygments (syntax highlighting) style to use.
pygments_style = 'sphinx'
# A list of ignored prefixes for module index sorting.
#modindex_common_prefix = []
# If true, keep warnings as "system message" paragraphs in the built documents.
#keep_warnings = False
# -- Options for HTML output ----------------------------------------------
# The theme to use for HTML and HTML Help pages. See the documentation for
# a list of builtin themes.
html_theme = 'sage-classic'
html_theme_path = ['../themes']
# Theme options are theme-specific and customize the look and feel of a theme
# further. For a list of options available for each theme, see the
# documentation.
html_theme_options = {}
# The name for this set of Sphinx documents. If None, it defaults to
# "<project> v<release> documentation".
# A shorter title for the navigation bar. Default is the same as html_title.
#html_short_title = None
# The name of an image file (relative to this directory) to place at the top
# of the sidebar.
#html_logo = None
# The name of an image file (within the static path) to use as favicon of the
# docs. This file should be a Windows icon file (.ico) being 16x16 or 32x32
# pixels large.
#html_favicon = None
# Add any paths that contain custom static files (such as style sheets) here,
# relative to this directory. They are copied after the builtin static files,
# so a file named "default.css" will overwrite the builtin "default.css".
html_static_path = [] #['_static']
# Add any extra paths that contain custom files (such as robots.txt or
# .htaccess) here, relative to this directory. These files are copied
# directly to the root of the documentation.
#html_extra_path = []
# If not '', a 'Last updated on:' timestamp is inserted at every page bottom,
# using the given strftime format.
#html_last_updated_fmt = '%b %d, %Y'
# If true, SmartyPants will be used to convert quotes and dashes to
# typographically correct entities.
#html_use_smartypants = True
# Custom sidebar templates, maps document names to template names.
#html_sidebars = {}
# Additional templates that should be rendered to pages, maps page names to
# template names.
#html_additional_pages = {}
# If false, no module index is generated.
#html_domain_indices = True
# If false, no index is generated.
#html_use_index = True
# If true, the index is split into individual pages for each letter.
#html_split_index = False
# If true, links to the reST sources are added to the pages.
#html_show_sourcelink = True
# If true, "Created using Sphinx" is shown in the HTML footer. Default is True.
#html_show_sphinx = True
# If true, "(C) Copyright ..." is shown in the HTML footer. Default is True.
#html_show_copyright = True
# If true, an OpenSearch description file will be output, and all pages will
# contain a <link> tag referring to it. The value of this option must be the
# base URL from which the finished HTML is served.
#html_use_opensearch = ''
# This is the file name suffix for HTML files (e.g. ".xhtml").
#html_file_suffix = None
# Output file base name for HTML help builder.
htmlhelp_basename = package_name + "doc"
# -- Options for LaTeX output ---------------------------------------------
latex_elements = {
# The paper size ('letterpaper' or 'a4paper').
#'papersize': 'letterpaper',
# The font size ('10pt', '11pt' or '12pt').
#'pointsize': '10pt',
# Additional stuff for the LaTeX preamble.
'preamble': '',
}
# Grouping the document tree into LaTeX files. List of tuples
# (source start file, target name, title,
# author, documentclass [howto, manual, or own class]).
latex_documents = [
('index', package_name + '.tex', u'Documentation of ' + six.text_type(package_name),
authors, 'manual'),
]
# The name of an image file (relative to this directory) to place at the top of
# the title page.
#latex_logo = None
# For "manual" documents, if this is true, then toplevel headings are parts,
# not chapters.
#latex_use_parts = False
# If true, show page references after internal links.
#latex_show_pagerefs = False
# If true, show URL addresses after external links.
#latex_show_urls = False
# Documents to append as an appendix to all manuals.
#latex_appendices = []
# If false, no module index is generated.
#latex_domain_indices = True
# -- Options for manual page output ---------------------------------------
# One entry per manual page. List of tuples
# (source start file, name, description, authors, manual section).
man_pages = [
('index', package_name, six.text_type(package_name) + u" documentation",
[authors], 1)
]
# If true, show URL addresses after external links.
#man_show_urls = False
# -- Options for Texinfo output -------------------------------------------
# Grouping the document tree into Texinfo files. List of tuples
# (source start file, target name, title, author,
# dir menu entry, description, category)
texinfo_documents = [
('index', package_name, six.text_type(package_name) + u" documentation",
authors, package_name, project,
'Miscellaneous'),
]
# Documents to append as an appendix to all manuals.
#texinfo_appendices = []
# If false, no module index is generated.
#texinfo_domain_indices = True
# How to display URL addresses: 'footnote', 'no', or 'inline'.
#texinfo_show_urls = 'footnote'
# If true, do not generate a @detailmenu in the "Top" node's menu.
#texinfo_no_detailmenu = False
# -- Options copied from Sagemath conf.py file -------------------------------
# We use MathJax to build the documentation unless the environment
# variable SAGE_DOC_MATHJAX is set to "no" or "False". (Note that if
# the user does not set this variable, then the script sage-env sets
# it to "True".)
if (os.environ.get('SAGE_DOC_MATHJAX', 'no') != 'no'
and os.environ.get('SAGE_DOC_MATHJAX', 'no') != 'False'):
extensions.append('sphinx.ext.mathjax')
mathjax_path = 'MathJax.js?config=TeX-AMS_HTML-full,../mathjax_sage.js'
from sage.misc.latex_macros import sage_mathjax_macros
# this is broken for now
# html_theme_options['mathjax_macros'] = sage_mathjax_macros()
## from pkg_resources import Requirement, working_set
## sagenb_path = working_set.find(Requirement.parse('sagenb')).location
## mathjax_relative = os.path.join('sagenb','data','mathjax')
## # It would be really nice if sphinx would copy the entire mathjax directory,
## # (so we could have a _static/mathjax directory), rather than the contents of the directory
## mathjax_static = os.path.join(sagenb_path, mathjax_relative)
## html_static_path.append(mathjax_static)
## exclude_patterns=['**/'+os.path.join(mathjax_relative, i) for i in ('docs', 'README*', 'test',
## 'unpacked', 'LICENSE')]
from sage.env import SAGE_LOCAL, SAGE_SHARE
html_static_path.append(SAGE_LOCAL + "/lib/mathjax") # conda
html_static_path.append(SAGE_SHARE + "/mathjax") # sage distribution
else:
extensions.append('sphinx.ext.imgmath')
# This is to make the verbatim font smaller;
# Verbatim environment is not breaking long lines
from sphinx.highlighting import PygmentsBridge
from pygments.formatters.latex import LatexFormatter
class CustomLatexFormatter(LatexFormatter):
def __init__(self, **options):
super(CustomLatexFormatter, self).__init__(**options)
self.verboptions = r"formatcom=\footnotesize"
PygmentsBridge.latex_formatter = CustomLatexFormatter
latex_elements['preamble'] += r'''
% One-column index
\makeatletter
\renewenvironment{theindex}{
\chapter*{\indexname}
\markboth{\MakeUppercase\indexname}{\MakeUppercase\indexname}
\setlength{\parskip}{0.1em}
\relax
\let\item\@idxitem
}{}
\makeatother
\renewcommand{\ttdefault}{txtt}
'''
#####################################################
# add LaTeX macros for Sage
from sage.misc.latex_macros import sage_latex_macros
try:
pngmath_latex_preamble # check whether this is already defined
except NameError:
pngmath_latex_preamble = ""
for macro in sage_latex_macros():
# used when building latex and pdf versions
latex_elements['preamble'] += macro + '\n'
# used when building html version
pngmath_latex_preamble += macro + '\n'
## The following is needed on conda-forge sagemath
from sage.repl.user_globals import initialize_globals
import sage.all
my_globs = dict()
initialize_globals(sage.all, my_globs)

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.. nodoctest
Constructions
=============
.. automodule:: vector_bundle.constructions
:members:
:undoc-members:
:show-inheritance:

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.. nodoctest
Ext group
=========
.. automodule:: vector_bundle.ext_group
:members:
:undoc-members:
:show-inheritance:

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.. nodoctest
Hom bundle
==========
.. automodule:: vector_bundle.hom_bundle
:members:
:undoc-members:
:show-inheritance:

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==========================
Sage Vector Bundle Package
==========================
This is a SageMath package implementing vector bundles on algebraic curves
To use this module, you need to import it::
from vector_bundle import *
This work is licensed under a `Creative Commons Attribution-Share Alike
3.0 License`__.
The source code for building this documentation and its theme are taken from
the `sage_sample`__ github repository.
__ https://creativecommons.org/licenses/by-sa/3.0/
__ https://github.com/sagemath/sage_sample
Vector Bundle
=============
.. toctree::
:maxdepth: 1
vector_bundle
hom_bundle
ext_group
constructions
Indices and Tables
==================
* :ref:`genindex`
* :ref:`modindex`
* :ref:`search`

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.. nodoctest
Vector bundle
=================
.. automodule:: vector_bundle.vector_bundle
:members:
:undoc-members:
:show-inheritance:

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{#
This is a customized version for Sage documentation theme.
Changes: at line 42, "slice(2)" -> "slice(1)"
#}
{#
basic/genindex-single.html
~~~~~~~~~~~~~~~~~~~~~~~~~~
Template for a "single" page of a split index.
:copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS.
:license: BSD, see LICENSE for details.
#}
{% macro indexentries(firstname, links) %}
{%- if links -%}
<a href="{{ links[0][1] }}">
{%- if links[0][0] %}<strong>{% endif -%}
{{ firstname|e }}
{%- if links[0][0] %}</strong>{% endif -%}
</a>
{%- for ismain, link in links[1:] -%}
, <a href="{{ link }}">{% if ismain %}<strong>{% endif -%}
[{{ loop.index }}]
{%- if ismain %}</strong>{% endif -%}
</a>
{%- endfor %}
{%- else %}
{{ firstname|e }}
{%- endif %}
{% endmacro %}
{%- extends "layout.html" %}
{% set title = _('Index') %}
{% block body %}
<h1 id="index">{% trans key=key %}Index &ndash; {{ key }}{% endtrans %}</h1>
<table style="width: 100%" class="indextable"><tr>
{%- for column in entries|slice(1) if column %}
<td style="width: 33%; vertical-align: top;"><ul>
{%- for entryname, (links, subitems, _) in column %}
<li>{{ indexentries(entryname, links) }}
{%- if subitems %}
<ul>
{%- for subentryname, subentrylinks in subitems %}
<li>{{ indexentries(subentryname, subentrylinks) }}</li>
{%- endfor %}
</ul>
{%- endif -%}</li>
{%- endfor %}
</ul></td>
{%- endfor %}
</tr></table>
{% endblock %}
{% block sidebarrel %}
<h4>{{ _('Index') }}</h4>
<p>{% for key, dummy in genindexentries -%}
<a href="{{ pathto('genindex-' + key) }}"><strong>{{ key }}</strong></a>
{% if not loop.last %}| {% endif %}
{%- endfor %}</p>
<p><a href="{{ pathto('genindex-all') }}"><strong>{{ _('Full index on one page') }}</strong></a></p>
{{ super() }}
{% endblock %}

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{#
This is a customized version for Sage documentation theme.
Changes: no change
#}
{#
basic/genindex-split.html
~~~~~~~~~~~~~~~~~~~~~~~~~
Template for a "split" index overview page.
:copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS.
:license: BSD, see LICENSE for details.
#}
{%- extends "layout.html" %}
{% set title = _('Index') %}
{% block body %}
<h1 id="index">{{ _('Index') }}</h1>
<p>{{ _('Index pages by letter') }}:</p>
<div class="genindex-jumpbox">
<p>{% for key, dummy in genindexentries -%}
<a href="{{ pathto('genindex-' + key) }}"><strong>{{ key }}</strong></a>
{% if not loop.last %}| {% endif %}
{%- endfor %}</p>
<p><a href="{{ pathto('genindex-all') }}"><strong>{{ _('Full index on one page') }}</strong>
({{ _('can be huge') }})</a></p>
</div>
{% endblock %}
{% block sidebarrel %}
{% if split_index %}
<h4>Index</h4>
<p>{% for key, dummy in genindexentries -%}
<a href="{{ pathto('genindex-' + key) }}"><strong>{{ key }}</strong></a>
{% if not loop.last %}| {% endif %}
{%- endfor %}</p>
<p><a href="{{ pathto('genindex-all') }}"><strong>{{ _('Full index on one page') }}</strong></a></p>
{% endif %}
{{ super() }}
{% endblock %}

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{#
This is a customized version for Sage documentation theme.
Changes: at line 52, "slice(2)" -> "slice(1)"
#}
{#
basic/genindex.html
~~~~~~~~~~~~~~~~~~~
Template for an "all-in-one" index.
:copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS.
:license: BSD, see LICENSE for details.
#}
{%- extends "layout.html" %}
{% set title = _('Index') %}
{% macro indexentries(firstname, links) %}
{%- if links -%}
<a href="{{ links[0][1] }}">
{%- if links[0][0] %}<strong>{% endif -%}
{{ firstname|e }}
{%- if links[0][0] %}</strong>{% endif -%}
</a>
{%- for ismain, link in links[1:] -%}
, <a href="{{ link }}">{% if ismain %}<strong>{% endif -%}
[{{ loop.index }}]
{%- if ismain %}</strong>{% endif -%}
</a>
{%- endfor %}
{%- else %}
{{ firstname|e }}
{%- endif %}
{% endmacro %}
{% block body %}
<h1 id="index">{{ _('Index') }}</h1>
<div class="genindex-jumpbox">
{% for key, dummy in genindexentries -%}
<a href="#{{ key }}"><strong>{{ key }}</strong></a>
{% if not loop.last %}| {% endif %}
{%- endfor %}
</div>
{%- for key, entries in genindexentries %}
<h2 id="{{ key }}">{{ key }}</h2>
<table style="width: 100%" class="indextable genindextable"><tr>
{%- for column in entries|slice_index(1) if column %}
<td style="width: 33%; vertical-align: top;"><ul>
{%- for entryname, (links, subitems, _) in column %}
<li>{{ indexentries(entryname, links) }}
{%- if subitems %}
<ul>
{%- for subentryname, subentrylinks in subitems %}
<li>{{ indexentries(subentryname, subentrylinks) }}</li>
{%- endfor %}
</ul>
{%- endif -%}</li>
{%- endfor %}
</ul></td>
{%- endfor %}
</tr></table>
{% endfor %}
{% endblock %}
{% block sidebarrel %}
{% if split_index %}
<h4>{{ _('Index') }}</h4>
<p>{% for key, dummy in genindexentries -%}
<a href="{{ pathto('genindex-' + key) }}"><strong>{{ key }}</strong></a>
{% if not loop.last %}| {% endif %}
{%- endfor %}</p>
<p><a href="{{ pathto('genindex-all') }}"><strong>{{ _('Full index on one page') }}</strong></a></p>
{% endif %}
{{ super() }}
{% endblock %}

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{% extends "classic/layout.html" %}
{% block rootrellink %}
<li class="nav-item nav-item-0">
<a href="http://www.sagemath.org"><img src="{{ pathto('_static/logo_sagemath_black.svg', 1) }}" class="sage-logo" title="Sage Logo"></a>
{% if website %}
<a href="index.html">{{ documentation_title|e }}</a>{{ reldelim1 }}
{% else %}
<a href="{{ documentation_root }}">{{ documentation_title|e }}</a>{{ reldelim1 }}
{% if refsub %}
<a href="{{ reference_root }}">{{ reference_title|e }}</a>{{ reldelim1 }}
{% endif %}
<a href="{{ pathto(root_doc)|e }}">{{ shorttitle|e }}</a>{{ reldelim1 }}
{% endif %}
</li>
{% endblock %}
{% block extrahead %}
<link rel="icon" href="{{ pathto('_static/sageicon.png', 1) }}" type="image/x-icon" />
{% endblock %}
{%- block footer %}
{{ super() }}
<script type="text/javascript">
/*global jQuery, window */
/* Sphinx sidebar toggle. Putting this code at the end of the body
* enables the toggle for the live, static, and offline docs. Note:
* sage.misc.html.math_parse() eats jQuery's dollar-sign shortcut. */
var jq = jQuery;
jq(document).ready(function () {
var bar, bod, bg, fg, key, tog, wid_old, wid_new, get_state, set_state;
bod = jq('div.bodywrapper');
bar = jq('div.sphinxsidebar');
tog = jq('<div class="sphinxsidebartoggle"></div>');
/* The sidebar toggle adapts its height to the bodywrapper height. */
const resizeObserver = new ResizeObserver(entries => {
tog.height(bod.height());
});
resizeObserver.observe(bod[0]);
/* Setup and add the toggle. See Sphinx v0.5.1 default.css. */
fg = jq('div.sphinxsidebar p a').css('color') || 'rgb(152, 219, 204)';
bg = jq('div.document').css('background-color') || 'rgb(28, 78, 99)';
wid_old = '230px';
wid_new = '5px';
tog.css('background-color', bg)
.css('border-width', '0px')
.css('border-right', wid_new + ' ridge ' + bg)
.css('cursor', 'pointer')
.css('position', 'absolute')
.css('left', '-' + wid_new)
.css('top', '0px')
.css('width', wid_new);
bod.css('position', 'relative');
bod.prepend(tog);
/* Cookie helpers. */
key = 'sphinxsidebar=';
set_state = function (s) {
var date = new Date();
/* Expiry in 7 days. */
date.setTime(date.getTime() + (7 * 24 * 3600 * 1000));
document.cookie = key + encodeURIComponent(s) + '; expires=' +
date.toUTCString() + '; path=/';
};
get_state = function () {
var i, c, crumbs = document.cookie.split(';');
for (i = 0; i < crumbs.length; i += 1) {
c = crumbs[i].replace(/^\s+/, '');
if (c.indexOf(key) === 0) {
return decodeURIComponent(c.substring(key.length, c.length));
}
}
return null;
};
/* Event handlers. */
tog.mouseover(function (ev) {
tog.css('border-right-color', fg);
}).mouseout(function (ev) {
tog.css('border-right-color', bg);
}).click(function (ev) {
if (bod.hasClass('wide')) {
bod.removeClass('wide');
bod.css('margin-left', wid_old);
bar.css('width', wid_old);
bar.show();
set_state('visible');
} else {
set_state('hidden');
bar.hide();
bar.css('width', '0px');
bod.css('margin-left', wid_new);
bod.addClass('wide');
}
});
/* Hide the normally visible sidebar? */
if (get_state() === 'hidden') {
tog.trigger('click');
} else {
set_state('visible');
}
});
</script>
<script type="text/javascript">
/* detex the document title by removing "\(", "\)", "\", "$" */
document.title = document.title.replace(/\\\(/g, '').replace(/\\\)/g, '').replace(/\\/g, '').replace(/\$/g, '');
</script>
{%- endblock %}
<!-- This macro block for the sidebar is heavily borrowed from the basic -->
<!-- theme of Sphinx. In particular, we borrowed from the file -->
<!-- themes/basic/layout.html distributed with Sphinx. -->
{%- macro sidebar() %}
{%- if not embedded %}{% if not theme_nosidebar|tobool %}
<div class="sphinxsidebar">
<div class="sphinxsidebarwrapper">
{%- block sidebarlogo %}
{%- if logo %}
<p class="logo"><a href="{{ pathto(master_doc) }}">
<img class="logo" src="{{ pathto('_static/' + logo, 1) }}" alt="Logo"/>
</a></p>
{%- endif %}
{%- endblock %}
{%- block sidebartoc %}
{%- if display_toc %}
<h3><a href="{{ pathto(master_doc) }}">{{ _('Table Of Contents') }}</a></h3>
{{ toc }}
{%- endif %}
{%- endblock %}
{%- block sidebarrel %}
{%- if prev %}
<h4>{{ _('Previous topic') }}</h4>
<p class="topless"><a href="{{ prev.link|e }}"
title="{{ _('previous chapter') }}">{{ prev.title }}</a></p>
{%- endif %}
{%- if next %}
<h4>{{ _('Next topic') }}</h4>
<p class="topless"><a href="{{ next.link|e }}"
title="{{ _('next chapter') }}">{{ next.title }}</a></p>
{%- endif %}
{%- endblock %}
{%- block sidebarsourcelink %}
{%- if show_source and has_source and sourcename %}
<h3>{{ _('This Page') }}</h3>
<ul class="this-page-menu">
<li><a href="{{ pathto('_sources/' + sourcename, true)|e }}"
rel="nofollow">{{ _('Show Source') }}</a></li>
</ul>
{%- endif %}
{%- endblock %}
{%- if customsidebar %}
{% include customsidebar %}
{%- endif %}
{%- block sidebarsearch %}
{%- if pagename != "search" and builder != "singlehtml" %}
<div id="searchbox" style="display: none" role="search">
<h3 id="searchlabel">{{ _('Quick search') }}</h3>
<div class="searchformwrapper">
<form class="search" action="{{ pathto('search') }}" method="get">
<input type="text" name="q" aria-labelledby="searchlabel" autocomplete="off" autocorrect="off" autocapitalize="off" spellcheck="false"/>
<!-- The shading of the "Go" button should be consistent -->
<!-- with the colour of the header and footer. See the file -->
<!-- doc/common/themes/sage/theme.conf for colours used by -->
<!-- the Sage theme. -->
<input type="submit" style="background-color: #B8B9F6" value="{{ _('Go') }}" />
</form>
<p class="searchtip" style="font-size: 90%">
{{ _('Enter search terms or a module, class or function name.') }}
</p>
</div>
</div>
<script>$('#searchbox').show(0);</script>
{%- endif %}
{%- endblock %}
</div>
</div>
{%- endif %}{% endif %}
{%- endmacro %}

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{#
basic/search.html
~~~~~~~~~~~~~~~~~
Template for the search page.
:copyright: Copyright 2007-2021 by the Sphinx team, see AUTHORS.
:license: BSD, see LICENSE for details.
#}
{%- extends "layout.html" %}
{% set title = _('Search') %}
{%- block scripts %}
{{ super() }}
<script src="{{ pathto('_static/searchtools.js', 1) }}"></script>
<script src="{{ pathto('_static/language_data.js', 1) }}"></script>
{%- endblock %}
{% block extrahead %}
<script src="{{ pathto('searchindex.js', 1) }}" defer></script>
{{ super() }}
{% endblock %}
{% block body %}
<h1 id="search-documentation">{{ _('Search') }}</h1>
{% block scriptwarning %}
<noscript>
<div class="admonition warning">
<p>
{% trans %}Please activate JavaScript to enable the search
functionality.{% endtrans %}
</p>
</div>
</noscript>
{% endblock %}
{% block searchtext %}
<p>
{% trans %}Searching for multiple words only shows matches that contain
all words.{% endtrans %}
</p>
<p>
{% trans %}Note also that you can also call "search_src(...)"
while running Sage to search Sage's source code.{% endtrans %}
</p>
{% endblock %}
{% block searchbox %}
<form action="" method="get">
<input type="text" name="q" aria-labelledby="search-documentation" value="" autocomplete="off" autocorrect="off" autocapitalize="off" spellcheck="false"/>
<input type="submit" value="{{ _('search') }}" />
<span id="search-progress" style="padding-left: 10px"></span>
</form>
{% endblock %}
{% block searchresults %}
{% if search_performed %}
<h2>{{ _('Search Results') }}</h2>
{% if not search_results %}
<p>{{ _('Your search did not match any documents.') }}</p>
{% endif %}
{% endif %}
<div id="search-results">
{% if search_results %}
<ul>
{% for href, caption, context in search_results %}
<li><a href="{{ pathto(item.href) }}">{{ caption }}</a>
<div class="context">{{ context|e }}</div>
</li>
{% endfor %}
</ul>
{% endif %}
</div>
{% endblock %}
{% endblock %}

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/*
* This style sheet is borrowed from the Sphinx classic theme style sheet
* distributed as the file themes/classic/static/classic.css_t in Sphinx 4.4.0.
*
* Customizations for Sage follow from the end of the original style sheet.
*
*/
@import url("basic.css");
/* -- page layout ----------------------------------------------------------- */
html {
/* CSS hack for macOS's scrollbar */
background-color: #FFFFFF;
}
body {
font-family: {{ theme_bodyfont }};
font-size: 100%;
background-color: {{ theme_footerbgcolor }};
color: #000;
margin: 0;
padding: 0;
}
div.document {
background-color: {{ theme_sidebarbgcolor }};
}
div.documentwrapper {
float: left;
width: 100%;
}
div.bodywrapper {
margin: 0 0 0 {{ theme_sidebarwidth|todim }};
}
div.body {
background-color: {{ theme_bgcolor }};
color: {{ theme_textcolor }};
padding: 0 20px 30px 20px;
}
{%- if theme_rightsidebar|tobool %}
div.bodywrapper {
margin: 0 {{ theme_sidebarwidth|todim }} 0 0;
}
{%- endif %}
div.footer {
color: {{ theme_footertextcolor }};
width: 100%;
padding: 9px 0 9px 0;
text-align: center;
font-size: 75%;
}
div.footer a {
color: {{ theme_footertextcolor }};
text-decoration: underline;
}
div.related {
background-color: {{ theme_relbarbgcolor }};
line-height: 30px;
color: {{ theme_relbartextcolor }};
}
div.related a {
color: {{ theme_relbarlinkcolor }};
}
div.sphinxsidebar {
{%- if theme_stickysidebar|tobool %}
top: 30px;
bottom: 0;
margin: 0;
position: fixed;
overflow: auto;
height: auto;
{%- endif %}
{%- if theme_rightsidebar|tobool %}
float: right;
{%- if theme_stickysidebar|tobool %}
right: 0;
{%- endif %}
{%- endif %}
}
{%- if theme_stickysidebar|tobool %}
/* this is nice, but it leads to hidden headings when jumping
to an anchor */
/*
div.related {
position: fixed;
}
div.documentwrapper {
margin-top: 30px;
}
*/
{%- endif %}
div.sphinxsidebar h3 {
font-family: {{ theme_headfont }};
color: {{ theme_sidebartextcolor }};
font-size: 1.4em;
font-weight: normal;
margin: 0;
padding: 0;
}
div.sphinxsidebar h3 a {
color: {{ theme_sidebartextcolor }};
}
div.sphinxsidebar h4 {
font-family: {{ theme_headfont }};
color: {{ theme_sidebartextcolor }};
font-size: 1.3em;
font-weight: normal;
margin: 5px 0 0 0;
padding: 0;
}
div.sphinxsidebar p {
color: {{ theme_sidebartextcolor }};
}
div.sphinxsidebar p.topless {
margin: 5px 10px 10px 10px;
}
div.sphinxsidebar ul {
margin: 10px;
padding: 0;
color: {{ theme_sidebartextcolor }};
}
div.sphinxsidebar a {
color: {{ theme_sidebarlinkcolor }};
}
div.sphinxsidebar input {
border: 1px solid {{ theme_sidebarlinkcolor }};
font-family: sans-serif;
font-size: 1em;
}
{% if theme_collapsiblesidebar|tobool %}
/* for collapsible sidebar */
div#sidebarbutton {
background-color: {{ theme_sidebarbtncolor }};
}
{% endif %}
/* -- hyperlink styles ------------------------------------------------------ */
a {
color: {{ theme_linkcolor }};
text-decoration: none;
}
a:visited {
color: {{ theme_visitedlinkcolor }};
text-decoration: none;
}
a:hover {
text-decoration: underline;
}
{% if theme_externalrefs|tobool %}
a.external {
text-decoration: none;
border-bottom: 1px dashed {{ theme_linkcolor }};
}
a.external:hover {
text-decoration: none;
border-bottom: none;
}
a.external:visited {
text-decoration: none;
border-bottom: 1px dashed {{ theme_visitedlinkcolor }};
}
{% endif %}
/* -- body styles ----------------------------------------------------------- */
div.body h1,
div.body h2,
div.body h3,
div.body h4,
div.body h5,
div.body h6 {
font-family: {{ theme_headfont }};
background-color: {{ theme_headbgcolor }};
font-weight: normal;
color: {{ theme_headtextcolor }};
border-bottom: 1px solid #ccc;
margin: 20px -20px 10px -20px;
padding: 3px 0 3px 10px;
}
div.body h1 { margin-top: 0; font-size: 200%; }
div.body h2 { font-size: 160%; }
div.body h3 { font-size: 140%; }
div.body h4 { font-size: 120%; }
div.body h5 { font-size: 110%; }
div.body h6 { font-size: 100%; }
a.headerlink {
color: {{ theme_headlinkcolor }};
font-size: 0.8em;
padding: 0 4px 0 4px;
text-decoration: none;
}
a.headerlink:hover {
background-color: {{ theme_headlinkcolor }};
color: white;
}
div.body p, div.body dd, div.body li, div.body blockquote {
text-align: justify;
line-height: 130%;
}
div.admonition p.admonition-title + p {
display: inline;
}
div.admonition p {
margin-bottom: 5px;
}
div.admonition pre {
margin-bottom: 5px;
}
div.admonition ul, div.admonition ol {
margin-bottom: 5px;
}
div.note {
background-color: #eee;
border: 1px solid #ccc;
}
div.seealso {
background-color: #ffc;
border: 1px solid #ff6;
}
div.topic {
background-color: #eee;
}
div.warning {
background-color: #ffe4e4;
border: 1px solid #f66;
}
p.admonition-title {
display: inline;
}
p.admonition-title:after {
content: ":";
}
pre {
padding: 5px;
background-color: {{ theme_codebgcolor }};
color: {{ theme_codetextcolor }};
line-height: 120%;
border: 1px solid #ac9;
border-left: none;
border-right: none;
}
code {
background-color: #ecf0f3;
padding: 0 1px 0 1px;
font-size: 0.95em;
}
th, dl.field-list > dt {
background-color: #ede;
}
.warning code {
background: #efc2c2;
}
.note code {
background: #d6d6d6;
}
.viewcode-back {
font-family: {{ theme_bodyfont }};
}
div.viewcode-block:target {
background-color: #f4debf;
border-top: 1px solid #ac9;
border-bottom: 1px solid #ac9;
}
div.code-block-caption {
color: #efefef;
background-color: #1c4e63;
}
/*
* Customizations for Sage
*
*/
/* -- sage logo -------------------------- */
img.sage-logo {
height: 28px;
vertical-align: middle;
}
/* -- hyperlink styles ------------------- */
a,
a:visited {
color: {{ theme_linkcolor }};
text-decoration: none;
}
/* -- commands or code within text ------- */
tt {
background-color: #EAEAF8;
padding: 0 1px 0 1px;
font-size: 0.95em;
}
/* -- class and method headlines --------- */
dl.py.class > dt,
dl.py.method > dt {
overflow-x: scroll;
text-align: left;
}
dl.py.class > dd > p:first-child {
overflow-x: scroll;
white-space: nowrap;
}
/* -- search page ------------------------ */
ul.search li {
text-align: left;
}
/* -- references ------------------------- */
dl.citation p {
text-align: left;
}
/* -- jupyter-sphinx ------------------------------------------------------ */
#thebelab-activate-button {
position: fixed;
right: 10px;
bottom: 10px;
}
.jupyter_cell .thebelab-input {
display: none;
}

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[theme]
inherit = classic
stylesheet = sage.css
pygments_style = sphinx
[options]
# Background color for the footer line: dark blue
footerbgcolor = #8C8DE6
# Text color for the footer line: white
footertextcolor = #FFFFFF
# Background color for the sidebar: light bluish gray
sidebarbgcolor = #EAEAF8
# Text color for the sidebar: black
sidebartextcolor = #000000
# Link color for the sidebar: light dark blue
sidebarlinkcolor = #090999
# Background color for the relation bar: light grayish blue
relbarbgcolor = #B8B9F6
# Text color for the relation bar: light dark blue
relbartextcolor = #090999
# Link color for the relation bar: light dark blue
relbarlinkcolor = #090999
# Body text color: black
textcolor = #000000
# Background color for headings: light gray
headbgcolor = #F2F2F2
# Body link color: dark greenish blue
linkcolor = #45529B
# Background color for code blocks: very pale yellow
codebgcolor = #FFFFE5

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# This Makefile is for convenience as a reminder and shortcut for the most used commands
# Package folder
PACKAGE = vector_bundle
# change to your sage command if needed
SAGE = sage
all: install test
install:
$(SAGE) -pip install --upgrade -v .
uninstall:
$(SAGE) -pip uninstall $(PACKAGE)
develop:
$(SAGE) -pip install --upgrade -e .
test:
$(SAGE) setup.py test
coverage:
$(SAGE) -coverage $(PACKAGE)/*
doc:
cd docs && $(SAGE) -sh -c "make html"
doc-pdf:
cd docs && $(SAGE) -sh -c "make latexpdf"
clean: clean-doc
clean-doc:
cd docs && $(SAGE) -sh -c "make clean"
.PHONY: all install develop test coverage clean clean-doc doc doc-pdf

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## -*- encoding: utf-8 -*-
import os
import sys
from setuptools import setup
from codecs import open # To open the README file with proper encoding
from setuptools.command.test import test as TestCommand # for tests
# Get information from separate files (README, VERSION)
def readfile(filename):
with open(filename, encoding='utf-8') as f:
return f.read()
# For the tests
class SageTest(TestCommand):
def run_tests(self):
errno = os.system("sage -t --force-lib vector_bundle")
if errno != 0:
sys.exit(1)
setup(
name = "vector_bundle",
version = readfile("VERSION").strip(), # the VERSION file is shared with the documentation
description='A sage package implementing vector bundles on algebraic curves using only function fields',
long_description = readfile("README.rst"), # get the long description from the README
# For a Markdown README replace the above line by the following two lines:
# long_description = readfile("README.md"),
# long_description_content_type="text/markdown",
# url='https://github.com/sagemath/sage_sample',
author='Mickaël Montessinos',
author_email='mickael.montessinos@mif.vu.lt', # choose a main contact email
license='GPLv2+', # This should be consistent with the LICENCE file
classifiers=[
# How mature is this project? Common values are
# 3 - Alpha
# 4 - Beta
# 5 - Production/Stable
'Development Status :: 3 - Alpha',
'Intended Audience :: Science/Research',
'Topic :: Software Development :: Build Tools',
'Topic :: Scientific/Engineering :: Mathematics',
'License :: OSI Approved :: GNU General Public License v2 or later (GPLv2+)',
'Programming Language :: Python :: 3.7',
], # classifiers list: https://pypi.python.org/pypi?%3Aaction=list_classifiers
keywords = "Algebraic Geometry Number Theory Curves Vector Bundles",
packages = ['vector_bundle'],
cmdclass = {'test': SageTest}, # adding a special setup command for tests
setup_requires = ['sage-package'],
install_requires = ['sage-package', 'sphinx'],
)

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from __future__ import absolute_import
# Add the import for which you want to give a direct access
from .vector_bundle import VectorBundle
from .constructions import *

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r"""
This module provides functions for various interesting constructions of vector bundles.
AUTHORS:
_Mickaël Montessinos: initial implementation
"""
###########################################################################
# Copyright (C) 2024 Mickaël Montessinos (mickael.montessinos@mif.vu.lt),#
# #
# Distributed under the terms of the GNU General Public License (GPL) #
# either version 3, or (at your option) any later version #
# #
# http://www.gnu.org/licenses/ #
###########################################################################
from sage.matrix.constructor import matrix
from vector_bundle import VectorBundle
from . import function_field_utility
def trivial_bundle(K):
r"""
Return the structure sheaf of the algebraic curve with function field K.
EXAMPLES ::
sage: from vector_bundle import trivial_bundle
sage: F.<x> = FunctionField(GF(3))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 + x + 2)
sage: O = K.maximal_order()
sage: V = trivial_bundle(K)
sage: V._ideals
[Ideal (1) of Maximal order of Function field in y defined by y^2 + x + 2]
sage: V._g_finite
[1]
sage: V._g_infinite
[1]
"""
return VectorBundle(K, K.one().divisor())
def canonical_bundle(K):
r"""
Return a canonical line bundle over K suitable for explicit Serre duality.
EXAMPLES ::
sage: from vector_bundle import canonical_bundle, trivial_bundle
sage: F.<x> = FunctionField(GF(3))
sage: canonical_bundle(F).degree()
-2
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 - x^3 - x)
sage: L = canonical_bundle(K); L
Vector bundle of rank 1 over Function field in y defined by y^2 + 2*x^3 + 2*x
sage: L == trivial_bundle(K)
True
"""
pi,_ = function_field_utility.safe_uniformizer(K)
return VectorBundle(K, pi.differential().divisor())
def _euclid(a,b):
r"""
The Euclidian algorithm in `N` but outputs intermediate steps.
"""
u1 = a
u2 = b
res = []
while u2 != 0:
q, r = u1.quo_rem(u2)
res.append((u1, u2, q, r))
u1 = u2
u2 = r
return res
def atiyah_bundle(field, rank, degree, base=None):
r"""
Return `\alpha_{r,d}(F_r \otimes base)` in the notation of Theorem 6 [At57]_
, where `r` is ``rank`` and `d` is ``degree``.
INPUT:
- ``field`` - FunctionField; of genus 1 with an infinite place of degree 1
- ``rank`` - integer
- ``degree`` - integer
- ``base`` - line bundle of degree 0 over field ; (default = ``trivial_bundle(field)``)
EXAMPLES ::
sage: from vector_bundle import atiyah_bundle
sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(11))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 - x^3 - x)
sage: base = VectorBundle(K, K.places_finite()[0].divisor())
sage: E = atiyah_bundle(K, 5, 3, base)
sage: E.rank()
5
sage: E.degree()
3
sage: E.hom(E).h0()
[
[1 0 0 0 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
]
"""
if base is None:
base = trivial_bundle(field)
if rank <= 0 :
raise ValueError('rank must be positive')
if field.genus() != 1:
raise ValueError('field must have genus 1')
if base.function_field() != field:
raise ValueError('base must have field as its function_field.')
if degree < 0:
return atiyah_bundle(field, rank, -degree, base).dual()
divisor = field.places_infinite()[0].divisor()
gcd = _euclid(rank, degree)
plan = [(i % 2,q) for i,(_, _, q, _) in enumerate(gcd)]
a, b = plan[-1]
plan[-1] = (a, b - 1)
starting_rank = gcd[-1][1]
result = trivial_bundle(field)
line_bundle = VectorBundle(field, divisor)
for _ in range(starting_rank - 1):
result = result.extension_by_global_sections()
result = result.tensor_product(line_bundle)
for move, reps in reversed(plan):
for _ in range(reps):
if move == 0:
result = result.extension_by_global_sections()
else:
result = result.tensor_product(line_bundle)
return result
def savin_bundle(field, rank, degree, line, line_1, line_2):
r"""
Return a weakly stable bundle over field of rank ``rank`` and degree
``degree``
ALGORITHM:
Section V of [Sav08]_
INPUT:
- ``field`` -- FunctionField: the base of the bundle. Must have genus at least 2.
- ``rank`` -- Integer: the rank of the output bundle
- ``degree`` -- Integer: the degree of the output bundle
- ``line`` -- VectorBundle: line bundle of degree ``degree//rank + 1`` plays the role of `F` in the algorithm
- ``line_1`` -- VectorBundle: line bundle of degree ``degree // rank`` plays the role of `F_1` in the algorithm
- ``line_2`` -- VectorBundle: line bundle of degree ``degree // rank`` plays the role of `F_2` in the algorithm
EXAMPLE ::
sage: from vector_bundle import VectorBundle, savin_bundle
sage: F.<x> = FunctionField(GF(11))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 - x^5 + x)
sage: line = VectorBundle(K, 3 * K.places_infinite()[0].divisor())
sage: line_1 = VectorBundle(K, 2 * K.places_finite()[0].divisor())
sage: line_2 = VectorBundle(K, 2 * K.places_finite()[1].divisor())
sage: E = savin_bundle(K, 3, 7, line, line_1, line_2)
sage: E.rank()
3
sage: E.degree()
7
"""
if degree < 0:
return savin_bundle(field, rank, -degree, line, line_1, line_2).dual()
q, r = degree.quo_rem(rank)
if line.rank() != 1 or line_1.rank() != 1 or line_2.rank() != 1:
raise ValueError('The input bundles must have rank 1')
if line_1.degree() != q or line_1.degree() != q or line.degree() != q+1:
raise ValueError('At least one of the input line bundles has'
+ 'invalid degree')
E = line_1
for _ in range(rank - r - 1):
E = line_2.non_trivial_extension(E)
for _ in range(r):
E = line.non_trivial_extension(E)
return E
def rank_2_trivial_determinant_semistable_bundle(ksi, ext=None):
r"""
Construct the semi-stable vector bundle of rank 2 and trivial determinant
defined by the extension of ``ksi`` by ``ksi.dual()`` and nonzero extension
class ``ext``.
The fact that this vector bundle is semi-stable is Lemma 5.1 in [NR69]_.
If ``ext`` is None, we default to a default nonzero extension class.
INPUT:
- ``ksi`` -- a degree 1 line bundle over a function field of genus at least 2
- ``ext`` -- an object representing a class of extensions of ``ksi`` by ``ksi.dual()``. (Default: None)
EXAMPLES ::
sage: from vector_bundle import VectorBundle, trivial_bundle, rank_2_trivial_determinant_semistable_bundle
sage: F.<x> = FunctionField(GF(11))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 - x^5 - 1)
sage: ksi = VectorBundle(K, K.places_finite()[0].divisor())
sage: V = rank_2_trivial_determinant_semistable_bundle(ksi)
sage: V.rank()
2
sage: V.determinant() == trivial_bundle(K)
True
"""
if ksi.rank() != 1:
raise ValueError('ksi must have rank one')
if ksi.degree() != 1:
raise ValueError('ksi must have degree one')
if ksi.function_field().genus() < 2:
raise ValueError('The function field of ksi must have genus at least 2')
ext_group = ksi.extension_group(ksi.dual())
return ext_group.extension(ext)

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r"""
This module implements the ExtGroup and ExtGroupElement classes used for
building extensions of vector bundles.
AUTHORS:
_Mickaël Montessinos: initial implementation
"""
###########################################################################
# Copyright (C) 2024 Mickaël Montessinos (mickael.montessinos@mif.vu.lt),#
# #
# Distributed under the terms of the GNU General Public License (GPL) #
# either version 3, or (at your option) any later version #
# #
# http://www.gnu.org/licenses/ #
###########################################################################
from sage.structure.sage_object import SageObject
from sage.structure.element import is_Matrix
from sage.matrix.special import block_matrix
from sage.modules.free_module_element import vector
class ExtGroup(SageObject):
r"""
The group of extensions of ``left`` by ``right``.
The group `Ext^1(\mathrm{left},\mathrm{right})` is
`H^1(\mathrm{left}^\vee \otimes \mathrm{right})`.
Its elements may be represented as matrices of infinite répartitions
lying in `M_{\mathrm{right}.rank(),\mathrm{left}.rank()}(R)` or as
vectors of length ``self.dim()`` with coefficients in the coefficient
field, representing linear forms on
`H^0(\omega \otimes \mathrm{right}^\vee \otimes \mathrm{left})`, where
`\omega` is the canonical bundle of the function field of ``left`` and
``right``
The two representations are related via Serre duality.
If ``precompute_basis`` is set to ``True``, a basis of répartition
matrices is computed. Its element represent the linear forms
``vector([0,...,0,1,0,...,0])``. Otherwise, linear forms are converted
to elements of the `H^1` on the fly. You should set ``precompute_basis``
to ``True`` only if you plan to create several extensions with this group.
INPUT:
- ``left`` -- VectorBundle
- ``right`` -- VectorBundle; must have the same function field as left
- ``precompute_basis`` -- boolean
EXAMPLES ::
sage: from vector_bundle import VectorBundle, trivial_bundle
sage: F.<x> = FunctionField(GF(3))
sage: triv = trivial_bundle(F)
sage: triv.extension_group(triv)
Extension group of Vector bundle of rank 1 over Rational function field in x over Finite Field of size 3 by Vector bundle of rank 1 over Rational function field in x over Finite Field of size 3.
"""
def __init__(self, left, right, precompute_basis=False):
if not left._function_field == right._function_field:
raise ValueError('left and right should have the same function'
+ 'field')
self._left = left
self._right = right
self._hom = left.hom(right)
self._ext_dual_basis, self._ext_dual_bundle = self._hom.h1_dual()
self._s = len(self._ext_dual_basis)
if precompute_basis:
self._compute_basis()
else:
self._basis = None
def __hash__(self):
return hash((self._left, self._right))
def __eq__(self, other):
return self._left == other._left and self._right == other._right
def _repr_(self):
return "Extension group of %s by %s." % (self._left, self._right)
def _compute_basis(self):
r"""
Compute the representation of basis elements of ``self`` as elements of the `H^1`
"""
if self._basis is None:
self._basis = [self._hom.h1_element(
vector([0] * i + [1] + [0] * (self._s-i-1)))
for i in range(self._s)]
def dim(self):
r"""
Return the dimension of the extension group as a vector space over the
base coefficient field
EXAMPLES ::
sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(11))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 - x^5 - 1)
sage: ksi = VectorBundle(K, K.places_finite()[0].divisor())
sage: ext = ksi.extension_group(ksi.dual()); ext.dim()
3
"""
return self._s
def left(self):
r"""
Return the left vector bundle of ``self``.
EXAMPLES ::
sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(3))
sage: L1 = VectorBundle(F, x.zeros()[0].divisor())
sage: L2 = VectorBundle(F, x.poles()[0].divisor())
sage: ext = L1.extension_group(L2); ext.left() == L1
True
"""
return self._left
def right(self):
r"""
Return the right vector bundle of ``self``.
EXAMPLES ::
sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(3))
sage: L1 = VectorBundle(F, x.zeros()[0].divisor())
sage: L2 = VectorBundle(F, x.poles()[0].divisor())
sage: ext = L1.extension_group(L2); ext.right() == L2
True
"""
return self._right
def dual_basis(self):
r"""
Return a basis of the dual of the `Ext^1` group. This is a basis of
`H^0(\omega \otimes \mathrm{right}^\vee \otimes \mathrm{left})`, where
`\omega` is a canonical line bundle.
EXAMPLES ::
sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(11))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 - x^5 - 1)
sage: ksi = VectorBundle(K, K.places_finite()[0].divisor())
sage: ext = ksi.extension_group(ksi.dual()); ext.dual_basis()
[[x^4/(x^5 + 6)], [x^5/(x^5 + 6)], [(x^2/(x^5 + 6))*y + 10*x^2/(x^5 + 6)]]
"""
return self._ext_dual_basis
def basis(self):
r"""Return a basis of `Ext^1` group. Its element are matrices of
infinite répartitions represented by field elements.
Computes and stores the basis if it was not precomputed yet.
EXAMPLES ::
sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(11))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 - x^5 - 1)
sage: ksi = VectorBundle(K, K.places_finite()[0].divisor())
sage: ext = ksi.extension_group(ksi.dual()); ext.basis()
[[(x^9/(x^10 + 2*x^5 + 1))*y], [(x^3/(x^5 + 1))*y], [x^6/(x^5 + 1)]]
"""
self._compute_basis()
return self._basis
def _extension_from_ext_element(self, ext):
r"""
Return the extension of ``self.left() by ``self.right()`` encoded by
``ext``.
``ext`` is a matrix of elements of ``self._function_field`` which
represents the constant value over the infinite places of a répartition
matrix with support at infinity representing an element of
`H^1(\mathrm{hom}(\mathrm{left},\mathrm{right}))`.
Such an element encodes an extension `V`:
`0 \to \mathrm{other} \to V \to \mathrm{self} \to 0`
INPUT:
- ``ext`` -- a matrix of dimension ``right.rank(),left.rank()``
"""
from vector_bundle import VectorBundle
function_field = self._left._function_field
ideals = self._right._ideals + self._left._ideals
g_finite = block_matrix([[self._right._g_finite, 0],
[0, self._left._g_finite]])
g_infinite = block_matrix([[self._right._g_infinite,
-ext * self._left._g_infinite],
[0, self._left._g_infinite]])
return VectorBundle(function_field, ideals, g_finite, g_infinite)
def _extension_from_linear_form(self, form):
if self._basis is None:
ext = self._hom.h1_element(form)
ext = self._ext_dual_bundle._vector_to_matrix(ext).transpose()
else:
ext = sum([coeff * e for coeff, e in zip(form, self._basis)])
return self._extension_from_ext_element(ext)
def extension(self, ext=None):
r"""
Return the extension of ``self.left()`` by ``self.right()`` encoded by
``ext``.
``ext`` can be a matrix of elements of ``self._function_field``
which represents the constant value over the infinite places of a
répartition matrix with support at infinity representing an element of
`H^1(\mathrm{hom}(\mathrm{left},\mathrm{right}))`.
Such an element encodes an extension `V`:
`0 \to \mathrm{other} \to V \to \mathrm{self} \to 0`
``ext`` can also be a vector of length `self.dim()` representing an
extension in the basis of the Ext vector space.
By default, ``ext`` is chosen as any non trivial extension.
EXAMPLES ::
sage: from vector_bundle import VectorBundle, trivial_bundle
sage: F.<x> = FunctionField(GF(11))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 - x^3 - x)
sage: triv = trivial_bundle(K)
sage: ext = triv.extension_group(triv)
sage: V = ext.extension()
sage: V.rank()
2
sage: V.determinant() == triv
True
sage: V.h0()
[(1, 0)]
sage: V.end().h0()
[
[0 1] [1 0]
[0 0], [0 1]
]
"""
if ext is None:
ext = vector([1] + [0]*(self._s-1))
if is_Matrix(ext):
return self._extension_from_ext_element(ext)
else:
return self._extension_from_linear_form(ext)

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###########################################################################
# Copyright (C) 2024 Mickaël Montessinos (mickael.montessinos@mif.vu.lt),#
# #
# Distributed under the terms of the GNU General Public License (GPL) #
# either version 3, or (at your option) any later version #
# #
# http://www.gnu.org/licenses/ #
###########################################################################
from sage.matrix.constructor import matrix
from sage.rings.infinity import Infinity
from copy import copy
from sage.misc.cachefunc import cached_function
from sage.misc.misc_c import prod
from sage.matrix.constructor import matrix
from sage.matrix.special import block_matrix, elementary_matrix,\
identity_matrix
from sage.rings.function_field.function_field_rational\
import RationalFunctionField
from sage.rings.function_field.order_rational\
import FunctionFieldMaximalOrderInfinite_rational
@cached_function
def all_infinite_places(K):
r"""
Return a list of the infinite places of K of all degrees
INPUT:
- ``K`` -- FunctionField
"""
if isinstance(K,RationalFunctionField):
return [K.gen().poles()[0]]
deg = K.degree()
return sum([K.places_infinite(degree = deg) for deg in range(1, deg + 1)],
[])
def infinite_valuation(a):
r"""
Returns the valuation -deg of an element of a rational function field
The degree method returns the "height" of the element.
EXAMPLES:
sage: from vector_bundle.function_field_utility import infinite_valuation
sage: F.<x> = FunctionField(GF(3))
sage: infinite_valuation(x**-1 + x**-2)
1
"""
if a == 0:
return Infinity
return a.denominator().degree() - a.numerator().degree()
def infinite_mod(a,i):
r"""
Returns a mod x**-i
EXAMPLES:
sage: from vector_bundle.function_field_utility import infinite_mod
sage: K.<x> = FunctionField(GF(3))
sage: infinite_mod(x**-1 + x**-3,2)
1/x
"""
x = a.parent().gen()
b = a * x**(i-1)
return x**(1-i) * (b.numerator() // b.denominator())
def infinite_integral_matrix(mat):
r"""
Return an matrix with coefficient in the infinite maximal order and its denominator.
INPUT:
- ``mat`` -- Matrix with coefficients in a rational function field K
OUTPUT:
- ``int_mat`` -- Matrix with coefficients in K.maximal_order_infinite()
- ``den`` -- Element of K.maximal_order_infinite such that mat = int_mat/den
EXAMPLES:
sage: from vector_bundle.function_field_utility import infinite_integral_matrix
sage: F.<x> = FunctionField(GF(11))
sage: mat = matrix([[x, 1], [x**-1, 2]])
sage: infinite_integral_matrix(mat)
(
[ 1 1/x]
[1/x^2 2/x], 1/x
)
"""
K = mat[0,0].parent()
if isinstance(K,FunctionFieldMaximalOrderInfinite_rational):
return mat,1
if not isinstance(K,RationalFunctionField):
raise ValueError('mat must have coefficients in a rational function'
+ 'field or its infinite maximal order')
x = K.gen()
R = K.maximal_order_infinite()
den = x**min([infinite_valuation(e) for e in mat.list()])
int_mat = matrix(R,mat.nrows(),mat.ncols(),(den*mat).list())
return int_mat, den
def infinite_hermite_form(mat,include_zero_cols=True,transformation=False):
r"""
Return the hermite form of a matrix with coefficient in a rational infinite maximal order.
EXAMPLE:
sage: from vector_bundle.function_field_utility import infinite_hermite_form
sage: K.<x> = FunctionField(GF(3))
sage: R = K.maximal_order_infinite()
sage: mat = matrix(R,[[1, x**-1, x**-2, (x**3+1) / x**3], [(2*x+2) / (x**3+2), x**-2, (x**2+2) / (x**4+1), 1]])
sage: H,T = infinite_hermite_form(mat,transformation=True); H
[0 0 1 0]
[0 0 0 1]
sage: mat*T == H
True
TESTS:
sage: F.<x> = FunctionField(GF(3))
sage: R = F.maximal_order_infinite()
sage: mat = matrix(R,[[x**-1, 0, 2, 1], [0, x**-1, 1, 1], [0, 0, 1, 0], [0, 0, 0, 1]])
sage: infinite_hermite_form(mat) == mat
True
"""
R = mat.base_ring()
if not isinstance(R,FunctionFieldMaximalOrderInfinite_rational):
raise ValueError('mat must have base ring a rational infinite maximal'
+ ' order.')
n = mat.nrows()
r = mat.ncols()
x = R.function_field().gen()
H = copy(mat)
T = identity_matrix(R,r)
#First, make mat upper triangular with diagonal coefficient of the form
#x**-k.
for i in range(1,n+1):
degs = [infinite_valuation(H[-i,j]) for j in range(r+1-i)]
d0 = min(degs)
j0 = degs.index(d0)
E = elementary_matrix(R,r,row1=j0,row2=r-i)
T *= E
H *= E
E = elementary_matrix(R,r,row1=r-i,scale=(x**d0 * H[-i,-i])**-1)
T *= E
H *= E
for j in range(r-i):
E = elementary_matrix(R,r,row1=r-i,row2=j,scale=-H[-i,j]/H[-i,-i])
T *= E
H *= E
for i in range(2,n+1):
d = infinite_valuation(H[-i,-i])
for j in range(1,i):
E = elementary_matrix(
R,r,row1=r-i,row2=r-j,
scale=(infinite_mod(H[-i,-j],d)-H[-i,-j])/H[-i,-i])
T *= E
H *= E
if not include_zero_cols:
H = H[:,r-n:]
if transformation:
return H,T
return H
def infinite_ideal_hnf(I,transformation=False):
r"""
Return the Hermite form of an ideal of the infinite maximal order.
"""
O = I.ring()
K = O.function_field()
x = K.gen()
F = K.base_field()
R = F.maximal_order_infinite()
n = K.degree()
order_basis = O.basis()
order_matrix = matrix(R,[gen.list() for gen in O.basis()]).transpose()
ideal_basis = I.gens_over_base();
ideal_matrix = order_matrix**-1 * matrix(F,[gen.list()
for gen in ideal_basis]).transpose()
mat,den = infinite_integral_matrix(ideal_matrix)
#This is awkward but if transformation is False, hnf,U = ().hermite_form()
#will unpack the matrix.
if transformation:
hnf,U = infinite_hermite_form(mat, transformation=True)
return hnf/den,U
hnf = infinite_hermite_form(mat)
return hnf/den
def infinite_order_xgcd(ideals):
r"""
Performs the extended gcd algorithm for ideals in the infinite order.
INPUT:
- ``ideals`` -- list of ideals over the infinite maximal order of a function field
OUTPUT:
- ``coeffs`` --- list of elements of the function field such that as[i] in ideals[i] and sum(as) = 1
ALGORITHM:
Proposition 1.3.7 from [Coh00]
EXAMPLES:
sage: from vector_bundle.function_field_utility import infinite_order_xgcd
sage: F.<x> = FunctionField(GF(3))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y**2 - x**-5 - 1)
sage: primes = [p.prime_ideal() for p in K.places_infinite()]; len(primes)
2
sage: a = infinite_order_xgcd(primes); a
[2*y + 2, y + 2]
sage: sum(a)
1
sage: all([a[i] in primes[i] for i in range(2)])
True
"""
order_basis = ideals[0].ring().basis()
if order_basis[0] != 1:
raise ValueError('The first element of the basis of the order should'
+ ' be 1.')
n = len(order_basis)
k = len(ideals)
y = ideals[0].ring().function_field().gen()
ideals_hnf = [infinite_ideal_hnf(I) for I in ideals]
ideals_bases = [[sum([order_basis[i]*mat[i,j] for i in range(n)])
for j in range(n)]
for mat in ideals_hnf]
C = block_matrix([ideals_hnf])
C, den = infinite_integral_matrix(C)
H,U = infinite_hermite_form(C, include_zero_cols=False, transformation=True)
if not (H/den).is_one():
raise ValueError("The ideals should be coprime.")
v = U[:,-n].list()
return [sum([ideals_bases[i][j]*v[n*i+j] for j in range(n)]) for i in range(k)]
def infinite_approximation(places,valuations,residues):
r"""
Return a in the function field of places such that
(a - residues[i]) has valuation at least valuations[i] at places[i].
INPUT:
- ``places`` -- list of FunctionFieldPlace. Infinite places only.
- ``valuations`` -- list of integers of same length as places.
- ``residues`` -- list of elements of the function field.
ALGORITHM:
Proposition 1.3.11 from [Coh00]
"""
if len(places) == 1:
return residues[0]
valuations = [max(0,val) for val in valuations]
primes = [place.prime_ideal() for place in places]
I = prod([prime**(val+1) for prime, val in zip(primes, valuations)])
ideals = [I * prime**(-val-1)
for prime, val in zip(primes, valuations)]
coefficients = infinite_order_xgcd(ideals)
return sum([c * res for c,res in zip(coefficients,residues)])
@cached_function
def safe_uniformizer(K):
r"""
Return a safe uniformizer and an infinite place of self._function_field
A uniformizer is safe if its valuation at other infinite places is 0.
EXAMPLES:
sage: from vector_bundle.function_field_utility import safe_uniformizer
sage: F.<x> = FunctionField(GF(3))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 - x**-5 - 1)
sage: places = K.places_infinite()
sage: pi, place = safe_uniformizer(K); pi
((2*x + 1)/x)*y + (2*x + 2)/x
sage: place == places[0]
True
sage: pi.valuation(place)
1
sage: pi.valuation(places[1])
0
TESTS:
sage: from vector_bundle.function_field_utility import all_infinite_places
sage: F.<x> = FunctionField(GF(3))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 + x + 2)
sage: places = K.places_infinite()
sage: pi, place = safe_uniformizer(K); pi
1/x*y
sage: place == places[0]
True
sage: pi.valuation(place)
1
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^4 + (2*x^2 + 2)/x^2)
sage: pi, _ = safe_uniformizer(K)
sage: [pi.valuation(place) for place in all_infinite_places(K)]
[1, 0, 0]
sage: safe_uniformizer(F)
(1/x, Place (1/x))
"""
places = all_infinite_places(K)
n = len(places)
return (infinite_approximation(
places,
[2] + ([1]*(n-1)),
[places[0].local_uniformizer()] + ([1]*(n-1))),
places[0])
def local_expansion(place,pi,f):
r"""
Return a function giving the i-th coefficient of the expansion of f.
This uses code from sage.rings.function_field.maps.FunctionFieldCompletion.
While somewhat redundant, it adds the possibility to chose the uniformizer
with respect to which the expansion is computed.
INPUT:
- ``place`` -- FunctionFieldPlace; the place at which to expand
- ``pi`` -- The uniformizer giving variable for the power series
- ``f`` -- The function to expand
OUTPUT:
- a function taking as input an integer i and returning the coefficient of degree i
EXAMPLES:
sage: from vector_bundle.function_field_utility import local_expansion
sage: from vector_bundle.function_field_utility import safe_uniformizer
sage: F.<x> = FunctionField(GF(3))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 - x**-5 - 1)
sage: pi, place = safe_uniformizer(K)
sage: f = 1 / (1-pi)
sage: exp = local_expansion(place, pi, f)
sage: all([exp(i) == 1 for i in range(20)])
True
"""
if f == 0:
return lambda i : 0
K = place.function_field()
der = K.higher_derivation()
k, _, to_k = place.residue_field()
val = f.valuation(place)
e = f * pi**(-val)
return lambda i : to_k(der._derive(e, i - val, pi)) if i >= val else 0
def residue(place,pi,f):
r"""
Return the residue of constant répartition f at place with respect
to local uniformizer pi.
"""
if pi.valuation(place) != 1:
raise ValueError('pi must be a local uniformizer at place')
k, _, _ = place.residue_field()
kc = place.function_field().constant_base_field()
exp = local_expansion(place,pi,f)
high_res = exp(-1)
return k.over(kc)(high_res).trace()
def invert_trace(field,base,target):
r"""
Find an element of trace 1 over base in field.
EXAMPLES:
sage: from vector_bundle.function_field_utility import invert_trace
sage: base = GF(9)
sage: field = GF(9**3)
sage: a = invert_trace(field, base, 1); a
2*z6^4 + 2*z6^3 + z6 + 1
sage: field.over(base)(a).trace()
1
"""
if field == base:
if target not in field:
raise ValueError('Since field = base, target should be an element'
+ ' of field')
return target
as_ext = field.over(base)
d = as_ext.degree(base)
t = as_ext.gen()
i = [(t**j).trace() != 0 for j in range(d)].index(True)
return field(target * t**i/((t**i).trace()))
def insert_row(mat,i,row):
r"""
Return matrix mat with row inserted in ith position.
EXAMPLES:
sage: from vector_bundle.function_field_utility import insert_row
sage: mat = matrix(GF(3), 2, 2, [1, 2, 2, 1])
sage: insert_row(mat, 1, [0, 1])
[1 2]
[0 1]
[2 1]
"""
return matrix([mat[j] for j in range(i)]
+ [row]
+ [mat[j] for j in range(i,mat.nrows())])
def norm(v):
r"""
Return the norm of vector v: the maximal degree of its coefficients.
Input:
- v -- vector with coefficients in a RationalFunctionField
EXAMPLES:
sage: from vector_bundle.function_field_utility import norm
sage: R.<x> = GF(3)[]
sage: v = vector([x^3 + 3 + 1, x^2])
sage: norm(v)
3
"""
return max([c.degree() for c in v.list()])
def smallest_norm_first(mat,i = 0,norms=[]):
r"""
Swap rows of M so that the i-th row has smaller norm than rows below.
INPUT:
``mat`` -- matrix with coefficients in a RationalFunctionField
``i`` -- integer (default: `0`)
EXAMPLES:
sage: from vector_bundle.function_field_utility import smallest_norm_first
sage: R.<x> = GF(3)[]
sage: mat = matrix([[1, 1], [x^2, x^3], [1, x]])
sage: smallest_norm_first(mat, 1)
[0, 1, 3]
sage: mat
[ 1 1]
[ 1 x]
[x^2 x^3]
"""
if norms == []:
norms = [norm(row) for row in mat]
j = norms[i:].index(min(norms[i:]))
mat.swap_rows(i,i+j)
n = norms[i]
norms[i] = norms[j+i]
norms[j+i] = n
return norms

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r"""
This module implements the HomBundle class, for vector bundles constructed
as homomorphism sheaves between two vector bundles.
The class inherits from the VectorBundle class, but sections, either local
or global, are displayed as matrices.
EXAMPLES ::
sage: from vector_bundle import VectorBundle, trivial_bundle, savin_bundle
sage: F.<x> = FunctionField(GF(3))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 - x^5 - 1)
We construct a vector bundle of rank 2 and degree 4::
sage: F = VectorBundle(K, 3 * K.places_infinite()[0].divisor())
sage: F1 = VectorBundle(K, 2 * K.places_finite()[0].divisor())
sage: F2 = VectorBundle(K, 2 * K.places_finite()[1].divisor())
sage: V = savin_bundle(K, 2, 4, F, F1, F2); V.h0()
[(1, 0), (2*x, 1)]
We construct the ``HomBundle`` from `\mathcal{O}_X^2` to ``V``. Its global
sections should represent linear maps from `k^2` to `H^0(V)`, where `k`
is the constant field of `K`::
sage: domain = trivial_bundle(K).direct_sum_repeat(2)
sage: hom_bundle = domain.hom(V); hom_bundle.h0()
[
[1 0] [0 1] [2*x 0] [ 0 2*x]
[0 0], [0 0], [ 1 0], [ 0 1]
]
"""
###########################################################################
# Copyright (C) 2024 Mickaël Montessinos (mickael.montessinos@mif.vu.lt),#
# #
# Distributed under the terms of the GNU General Public License (GPL) #
# either version 3, or (at your option) any later version #
# #
# http://www.gnu.org/licenses/ #
###########################################################################
from sage.matrix.constructor import matrix
from sage.modules.free_module_element import vector
from vector_bundle import VectorBundle
class HomBundle(VectorBundle):
r"""
Vector bundles representing homomorphism sheaves of vector bundles.
EXAMPLES ::
sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(3))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 + x + 2)
sage: ideals = [P.prime_ideal() for P in K.places_finite()[:2]]
sage: g_finite = matrix([[1, x], [2, y]])
sage: g_infinite = matrix([[x, 1], [2, y]])
sage: V1 = VectorBundle(K, ideals, g_finite, g_infinite)
sage: V2 = VectorBundle(K, K.places_infinite()[0].divisor())
sage: V = V1.hom(V2); V
Homomorphism bundle from Vector bundle of rank 2 over Function field in y defined by y^2 + x + 2 to Vector bundle of rank 1 over Function field in y defined by y^2 + x + 2
"""
def __init__(self,domain,codomain):
if (not isinstance(domain,VectorBundle) or
not isinstance(codomain,VectorBundle)):
raise TypeError
if domain._function_field != codomain._function_field:
raise ValueError
self._domain = domain
self._codomain = codomain
ideals = [ideal_domain**-1 * ideal_codomain
for ideal_domain in domain._ideals
for ideal_codomain in codomain._ideals]
g_finite = (domain._g_finite.transpose()**-1)\
.tensor_product(codomain._g_finite)
g_infinite = (domain._g_infinite.transpose()**-1)\
.tensor_product(codomain._g_infinite)
super().__init__(domain._function_field, ideals,g_finite,g_infinite)
def __hash__(self):
return hash((self._domain, self._codomain))
def __eq__(self,other):
return (super().__eq__(other)
and self._domain == other._domain
and self._codomain == other._codomain)
def _repr_(self):
return "Homomorphism bundle from %s to %s" % (
self._domain,
self._codomain,
)
def domain(self):
r"""
Return the domain of self
EXAMPLES ::
sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(3))
sage: L1 = VectorBundle(F, x.poles()[0].divisor())
sage: L2 = VectorBundle(F, x.zeros()[0].divisor())
sage: V = L1.hom(L2); V.domain() == L1
True
"""
return self._domain
def codomain(self):
r"""
Return the codomain of self
EXAMPLES ::
sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(3))
sage: L1 = VectorBundle(F, x.poles()[0].divisor())
sage: L2 = VectorBundle(F, x.zeros()[0].divisor())
sage: V = L1.hom(L2); V.codomain() == L2
True
"""
return self._codomain
def _vector_to_matrix(self,vec):
r"""
Return the matrix of the homomorphism encoded by vector vec.
"""
return matrix(self._domain.rank(),self._codomain.rank(),vec).transpose()
def _matrix_to_vector(self,mat):
r"""
Inverse operation of _vector_to_matrix()
"""
return vector(mat.transpose().list())
def basis_finite(self):
r"""
Return basis of the finite lattice of the hom bundle.
OUTPUT:
- The basis elements are represented as matrices.
EXAMPLES ::
sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(3))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 + x + 2)
sage: ideals = [P.prime_ideal() for P in K.places_finite()[:2]]
sage: g_finite = matrix([[1, x], [2, y]])
sage: g_infinite = matrix([[x, 1], [2, y]])
sage: V1 = VectorBundle(K, ideals, g_finite, g_infinite)
sage: O = K.maximal_order()
sage: V2 = VectorBundle(K, K.places_finite()[2].prime_ideal(),1,x^2)
sage: V = V1.hom(V2)
sage: V.basis_finite()
[[(x/(x^2 + x + 2))*y + (x + 2)/(x^2 + x + 2) (x/(x^2 + x + 2))*y + 2*x^2/(x^2 + x + 2)],
[(2/(x^2 + x + 2))*y + x/(x^2 + x + 2) (2/(x^2 + x + 2))*y + x/(x^2 + x + 2)]]
"""
basis = super().basis_finite()
return [self._vector_to_matrix(v) for v in basis]
def basis_infinite(self):
r"""
Return basis of the infinite lattice of the hom bundle.
OUTPUT:
- The basis elements are represented as matrices.
EXAMPLES ::
sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(3))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 + x + 2)
sage: ideals = [P.prime_ideal() for P in K.places_finite()[:2]]
sage: g_finite = matrix([[1, x], [2, y]])
sage: g_infinite = matrix([[x, 1],[2, y]])
sage: V1 = VectorBundle(K, ideals,g_finite,g_infinite)
sage: O = K.maximal_order()
sage: V2 = VectorBundle(K, K.places_finite()[2].prime_ideal(),1,x^2)
sage: V = V1.hom(V2)
sage: V.basis_infinite()
[[(x^2/(x^3 + 2*x^2 + 1))*y + (x^4 + 2*x^3)/(x^3 + 2*x^2 + 1) (x^3/(x^3 + 2*x^2 + 1))*y + 2*x^2/(x^3 + 2*x^2 + 1)],
[(2*x^3/(x^3 + 2*x^2 + 1))*y + x^2/(x^3 + 2*x^2 + 1) (2*x^4/(x^3 + 2*x^2 + 1))*y + x^3/(x^3 + 2*x^2 + 1)]]
"""
basis = super().basis_infinite()
return [self._vector_to_matrix(v) for v in basis]
def basis_local(self,place):
r"""
Return basis of the infinite lattice of the hom bundle.
OUTPUT:
- The basis elements are represented as matrices.
EXAMPLES ::
sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(3))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 + x + 2)
sage: ideals = [P.prime_ideal() for P in K.places_finite()[:2]]
sage: g_finite = matrix([[1,x],[2,y]])
sage: g_infinite = matrix([[x,1],[2,y]])
sage: V1 = VectorBundle(K, ideals,g_finite,g_infinite)
sage: O = K.maximal_order()
sage: V2 = VectorBundle(K, K.places_finite()[2].prime_ideal(),1,x^2)
sage: V = V1.hom(V2)
sage: place = K.places_finite()[0]
sage: V.basis_local(place)
[[(1/(x^2 + x + 2))*y + (x + 2)/(x^3 + x^2 + 2*x) (1/(x^2 + x + 2))*y + 2*x/(x^2 + x + 2)],
[(2/(x^2 + x + 2))*y + x/(x^2 + x + 2) (2/(x^2 + x + 2))*y + x/(x^2 + x + 2)]]
sage: all([all([(mat * g_finite)[0, j].valuation(place) >= (V2._ideals[0] * V1._ideals[j]**-1).divisor().valuation(place) for j in range(2)]) for mat in V.basis_local(place)])
True
"""
basis = super().basis_local(place)
return [self._vector_to_matrix(v) for v in basis]
def hom(self, other):
r"""
Return the Hom bundle from self to other.
If other is a vector bundle, this is the hom bundle from ``self._codomain``
to ``self._domain.tensor_product(other)``.
If other is also a hom bundle, this is the hom bundle from
``self.codomain().tensor_product(other.domain())``
to ``self.domain().tensor_product(other.codomain()``
"""
if isinstance(other, HomBundle):
return self._codomain.tensor_product(other._domain)\
.hom(self._domain.tensor_product(other._codomain))
return self._codomain.hom(self._domain.tensor_product(other))
def tensor_product(self,other):
r"""
Return the tensor product of a hom bundle and a vector bundle.
This is the same thing as
``self._domain.hom(self._codomain.tensor_product(other))``
"""
return self._domain.hom(self._codomain.tensor_product(other))
def conorm(self,K):
r"""
Return the conorm of a hom bundle.
It is the same thing as the hom bundle of the conorms of its domain and
codomain.
EXAMPLES ::
sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(3))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 + x + 2)
sage: ideals = [P.prime_ideal() for P in F.places_finite()[:2]]
sage: g_finite = matrix([[1, x], [2, x]])
sage: g_infinite = matrix([[x, 1], [2, x]])
sage: V1 = VectorBundle(F, ideals, g_finite, g_infinite)
sage: ideals = [P.prime_ideal() for P in F.places_finite()[1:3]]
sage: g_finite = matrix([[0, x], [1, 1/x]])
sage: g_infinite = matrix([[x, 2*x^2], [2, 1]])
sage: V2 = VectorBundle(F, ideals, g_finite, g_infinite)
sage: V1.conorm(K).hom(V2.conorm(K)) == V1.hom(V2).conorm(K)
True
"""
return self._domain.conorm(K).hom(self._codomain.conorm(K))
def h0(self):
r"""
Returns the 0th cohomology group of the hom bundle.
The global sections are output in matrix form, they are the global
homomorphisms from ``self._domain`` to ``self._codomain``.
EXAMPLES ::
sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(3))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 + x + 2)
sage: ideals = [P.prime_ideal() for P in K.places_finite()[:2]]
sage: g_finite = matrix([[1, x], [2, y]])
sage: g_infinite = matrix([[x, 1], [2, y]])
sage: V1 = VectorBundle(K, ideals, g_finite, g_infinite)
sage: O = K.maximal_order()
sage: V2 = VectorBundle(K, K.places_finite()[2].prime_ideal(),1,x^2)
sage: V = V1.hom(V2)
sage: h0 = V.h0(); len(h0) == V.degree() + (1-K.genus())*V.rank()
True
sage: all([all([(mat * g_finite)[0, j] in V2._ideals[0] * V1._ideals[j]**-1 for j in range(2)]) for mat in h0])
True
sage: O_infinity = K.maximal_order_infinite()
sage: all([all([a in O_infinity for a in (x**-2 * mat * g_infinite).list()]) for mat in h0])
True
"""
h0 = super().h0()
return [self._vector_to_matrix(v) for v in h0]

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