Sage 9.2 Release Tour
released October 24, 2020
Contents

Sage 9.2 Release Tour
 Python 3 transition completed
 Package upgrades
 Graphics
 Linear and multilinear algebra
 Polyhedral geometry
 Combinatorics
 Graph theory
 Commutative algebra
 Manifolds
 Algebra
 Improved Unicode support
 Configuration and build changes
 New development tools
 Cleaning
 Availability of Sage 9.2 and installation help
 More details
Python 3 transition completed
SageMath 9.0 was the first version of Sage running on Python 3 by default. SageMath 9.1 continued to support Python 2.
Support for Python 2 removed
Sage 9.2 has removed support for Python 2. The Sage library now makes use of Python language and library features that are only available in Python 3.6 or newer; and large amounts of compatibility code have been removed.
However, note that this is unrelated to the minimal requirements for a source installation of the Sage distribution: Sage 9.2 is still able to build on a system that only provides Python 2.x or Python 3.5 or older. In this case, the SageMath distribution builds its own copy of Python 3.
Support for Python 3.6, 3.8, and 3.9 added
Sage 9.2 has added support for Python 3.8 in #27754 and Python 3.9 in #30184.
Sage 9.2 has also added support for Python 3.6. This allows Sage to use the system Python on some older Linux distributions that are still in widespread use in scientific computing, including centos8 and fedora{26,27,28} (although Python 3.7.x packages are also available for these). See #29033 for more details.
Hence, Sage 9.2 conforms to (and exceeds) NumPy Enhancement Proposal 29 regarding Python version support policies.
If no suitable system Python, versions 3.6.x, 3.7.x, 3.8.x, or 3.9.x is found, Sage installs its own copy of Python 3 from source. The version of Python shipped with the Sage distribution has been upgraded from 3.7.3 to 3.8.5.
For developers: Using Python 3.6+ features in sagelib
Metaticket #29756 provides a starting point for a discussion of new features of the Python language and standard library to bring them to systematic use in sagelib. All features provided by Python 3.6 can be used immediately; features introduced in Python 3.7 or later will require backporting or a decision to drop the goal of supporting Python 3.6.
More details
Trac tickets with keyword/component python3 in milestone 9.2
See Python3Switch for more details.
Package upgrades
The removal of support for Python 2 has enabled major package upgrades.
Major uservisible package upgrades below...
matplotlib
Dropping Python 2 support allowed us to make a major jump from matplotlib 2.2.5 to 3.3.1. See matplotlib's release notes for 3.0, 3.1, 3.2,3.3. In addition to improved output, this update will likely enable Sage developers to implement new features for plotting and graphics.
rpy2 and R
The rpy2 Python package is the foundation for SageMath's interface to R. Dropping Python 2 support allowed us to make the major upgrade from 2.8.2 to 3.3.5 in #29441; see the release notes for details.
We only did a minor upgrade of R itself in the Sage distribution, to 3.6.3, the latest in the 3.6.x series. Of course, if R 4.0.x is installed in the system, Sage will use it instead of building its own copy.
The SageMath developers are eager to learn from users how they use the SageMathR interface, and what needs to be added to it to become more powerful. Let us know at sagedevel.
sphinx
Sage uses Sphinx to build its documentation. Sage 9.2 has updated Sphinx from 1.8.5 to 3.1.2; see Sphinx release notes for more information.
SymPy
SymPy has been updated from 1.5 to 1.6.2 in #29730, #30425. See the Release notes.
IPython, Jupyter notebook, JupyterLab
Dropping support for Python 2 allowed us to upgrade IPython from 5.8.0 to 7.13.0 in #28197. See the release notes for the 6.x and 7.x series.
We have also upgraded the Jupyter notebook from 5.7.6 to 6.1.1 in #26919; see the notebook changelog for more information. Besides, the pdf export of Jupyter notebooks has been fixed, so that LaTeXtypeset outputs are now rendered in the pdf file (#23330).
JupyterLab is now fully supported as an optional, alternative interface #30246, including interacts. To use it, install it first, using the command sage i jupyterlab_widgets. Then you can start it using ./sage n jupyterlab.
Normaliz
The optional package Normaliz, a tool for computations in affine monoids, vector configurations, lattice polytopes, rational cones, and algebraic polyhedra has been upgraded from 3.7.2 to 3.8.8, and PyNormaliz to version 2.12.
The upgrade adds support for incremental ("dynamic") computations, the computation of automorphism groups and refined triangulations of cones and polyhedra, and limited support for semiopen cones and polyhedra.
To install Normaliz and PyNormaliz, use sage i pynormaliz.
SageTeX
Updated to version 3.5, improving Python 3 compatibility, also updated to version 3.5 on CTAN.
sws2rst + usage example
In ticket #28838, the command sage sws2rst was resurrected via a new pipinstallable package sagesws2rst. It can be installed in Sage 9.2 using sage i sage_sws2rst.
Below is an example of usage. First we download a sage worksheet (.sws) prepared for Sage Days 20 at CIRM (Marseille, 2010):
$ wget http://slabbe.org/Sage/2010perpignan/CIRM_Tutorial_1.sws $ ls CIRM_Tutorial_1.sws
We translate the sws worksheet into a ReStructuredText syntax file (.rst) using sage sws2rst. This creates also a directory of images:
$ sage sws2rst CIRM_Tutorial_1.sws Processing CIRM_Tutorial_1.sws File at CIRM_Tutorial_1.rst Image directory at CIRM_Tutorial_1_media $ ls CIRM_Tutorial_1_media CIRM_Tutorial_1.rst CIRM_Tutorial_1.sws
Then, we can check that it works properly by looking at the generated rst file. Alternatively, we can translate it to a basic html file using rst2html:
$ rst2html.py CIRM_Tutorial_1.rst CIRM_Tutorial_1.html CIRM_Tutorial_1.rst:176: (WARNING/2) Explicit markup ends without a blank line; unexpected unindent. CIRM_Tutorial_1.rst:334: (WARNING/2) Inline strong startstring without endstring.
As seen above, there are few warnings sometimes because the translation made by sws2rst is not 100% perfect, but most of it is okay:
$ firefox CIRM_Tutorial_1.html
Moreover, one can use the sage rst2ipynb script to translate the rst file obtained above to a Jupyter notebook:
$ sage rst2ipynb CIRM_Tutorial_1.rst CIRM_Tutorial_1.ipynb
One can check the result:
$ sage n jupyter
Note that to translate old .sws files to .ipynb, you may also use the export notebook:
$ sage n h [...] * List available legacy Sage notebooks: sage notebook=export list * Export a legacy Sage notebook as a Jupyter notebook: sage notebook=export ipynb=Output.ipynb admin:10
Other package updates
From the changelog of the release of 9.2, the following list of upgrades made in 9.2 was extracted:
#3360: sympow 2.023.6 (for GCC 10 support) #22191: ECL 20.4.24 #26891: Nauty 2.7 #26919: Jupyter notebook to latest (6.1.1) and its dependencies to latest #27309: FriCAS 1.3.6 #27754: Python 3.8.5 #27880: Kenzo and its interface #27952: Normaliz 3.8.8, PyNormaliz 2.12, add script package libnauty #28197: ipython 7 #28856: sphinx 3 #28959: zn_poly v0.9.2 #29061: symmetrica3.0.1 #29240: pexpect 4.8 #29313: pari 2.11.4 #29441: rpy2 package 2.8.2 > 3.3.5, Update R 3.6.3, add new dependencies #29480: Cython 0.29.17 #29483: gsl 2.6 #29547: matplotlib 3 #29552: giac 1.5.087 #29658: BRiAl 1.2.8 #29730: sympy 1.6 #29766: NumPy 1.19.1, scipy 1.5.2, networkx 2.4, add pybind11 package #29803: setuptools, setuptools_scm, pip (202006), add package wheel; remove zope_interface #29809: r117 #29826: eantic 0.1.7 #29859: palp 2.11 (for GCC 10 compatibility) #29861: Cython 0.29.21 #30001: sphinx 3.1 #30063: Maxima 5.44.0 #30150: cmake 3.18.2 #30176: matplotlib 3.3 #30185: pillow 7.2.0 #30262: eantic 0.1.8 #30317: pip 20.2.2, setuptools 49.6.0 #30338: libhomfly to the latest version #30342: sagetex 3.5 #30358: matplotlib 3.3.1, certifi 2020.6.20 #30412: gf2x 1.3 #30425: misc pip Update 202008: SymPy, pip, six #30583: gmpy2 2.1.0.b5 #30603: readline 8.0
For developers: Upgrading packages
Upgrading Python packages in the Sage distribution from PyPI has again become easier, thanks to #20104. You can now do:
$ sage package updatelatest matplotlib Updating matplotlib: 3.3.0 > 3.3.1 Downloading tarball to ...matplotlib3.3.1.tar.bz2 [...............................................................]
When you do this, please remember to check that the checksums.ini file has an upstream_url in the format upstream_url=https://pypi.io/packages/source/m/matplotlib/matplotlibVERSION.tar.gz. (This is not needed for updatelatest to work, but helps with automated tests of the upgrade ticket  see Sage 9.1 release tour on this topic.)
For packagers: Changes to packages
The packages giacpy_sage and sage_brial have been merged into sagelib as sage.libs.giac and sage.rings.polynomial.pbori.
The directory build/pkgs/sage_sws2rst/src contains a new pipinstallable package, providing the script sagesws2rst.
The Sage library is now built out of the directory build/pkgs/sagelib/src/. A pipinstallable source distribution (sdist) can be built using the script build/pkgs/sagelib/spkgsrc (#29411, #29950).
The scripts in src/bin/ are now installed by sagelib's setup.py (#21559). Also several scripts have been moved to build/bin/, and some obsolete scripts have been removed (#29825, #27171).
Many buildrelated functions of the main Sage script, src/bin/sage (installed as sage), have been moved to a script build/bin/sagesite (not installed) in #29111. It is hoped that downstream distribution packaging is able to use this cleaned up script instead of replacing it with an adhoc distributionspecific script  so that users can rely on a consistent interface. Contributions of further clean ups and refactoring of the script are welcome.
Graphics
New features
Specify the rectangle in which to draw a matrix using the new xrange and yrange options of matrix_plot. For example, to draw a matrix in [0,1]×[0,1] instead of the default [0.5,4.5]×[0.5,4.5]: matrix_plot(identity_matrix(5), xrange=(0, 1), yrange=(0, 1)). 27895 (Markus Wageringel)
Set the initial camera orientation in Three.js plots using the new viewpoint option. Pass it a list/tuple of the form [[x,y,z],angle], such as that provided by the existing Get Viewpoint option accessible from the menu button in the lowerright corner of a Three.js plot. 29192 (Paul Masson)
 Change the size, font, and opacity of text displayed in the Three.js viewer. For example:
30614 (Joshua Campbell)
Save a 3D graphics object directly to an HTML file that uses the Three.js viewer, similar to how you would save a PNG image: G.save('plot.html'). 29194 (Joshua Campbell)
Produce an interactive 3D animation that you can pan, rotate, and zoom while the animation is playing using the Three.js viewer. A slider and buttons for controlling playback are included on the page by default. To use this new feature construct an animation as you normally would, passing a list of still frames to the animate function, then call the interactive method. For example:
1 def build_frame(t): 2 """Build a single frame of animation at time t.""" 3 e = parametric_plot3d([sin(xt), 0, x], 4 (x, 0, 2*pi), color='red') 5 b = parametric_plot3d([0, sin(xt), x], 6 (x, 0, 2*pi), color='green') 7 return e + b 8 9 frames = [build_frame(t) 10 for t in (0, pi/32, pi/16, .., 2*pi)] 11 animate(frames, delay=5).interactive( 12 projection='orthographic')
29194 (Joshua Campbell)
Implementation improvements
Points are now sampled exponentially when scale='semilogx' or scale='loglog' is specified. This decreases the number of points necessary for an accurate plot (and also increases the chance that the default number of points will produce an acceptable plot). 29523 (Blair Mason)
Points and lines are now ignored in STL 3D export. Moreover disjoint union of surfaces can be saved. 29732 (Frédéric Chapoton)
Three.js has been upgraded to version r117. 29809 (Paul Masson)
Long text is no longer clipped in Three.js plots. Multiline text is not yet supported but is in the works. 29758 (Joshua Campbell)
JSmol's telemetry functionality has been disabled. It will no longer phone home when, for example, using viewer='jmol' in a Jupyter notebook. 30030 (Joshua Campbell)
 SVG export has been added to the javascript graph display tool:
G.show(method='js') 29807
For developers
Clarified that example Three.js plots in the documentation should use the online=True viewing option. 30136 (Paul Masson)
Linear and multilinear algebra
One free module constructor to rule them all
Sage has several specialized implementation classes for free modules and vector spaces. The factory functions FreeModule and VectorSpace select the appropriate class depending on the base ring and other parameters:
The free modules (vector spaces) created here have a distinguished standard basis indexed by range(rank).
In Sage 9.2, these factory functions have been extended in #30194 so that they cover two more cases:
1. If a sequence or family of indices is passed instead of the rank (dimension), then a CombinatorialFreeModule is created instead. These modules underly SageMath's facilities for algebraic combinatorics.
1 sage: U = FreeModule(AA, ['x', 'y', 'z']); U
2 Free module generated by {'x', 'y', 'z'} over Algebraic Real Field
3 sage: V = VectorSpace(QQ, ZZ); V
4 sage: V.basis()
5 Lazy family
6 (Term map from Integer Ring
7 to Free module generated by Integer Ring over Rational Field(i))
8 _{i in Integer Ring}
9 sage: QQ^SymmetricGroup(4)
10 Free module generated by
11 Symmetric group of order 4! as a permutation group over Rational Field
2. If the factory function is invoked with the parameter with_basis=None, then a free module of the given rank without distinguished basis is created.
1 sage: W = FreeModule(AA, 3, with_basis=None); W
2 3dimensional vector space over the Algebraic Real Field
3 sage: W.category()
4 Category of finite dimensional vector spaces over Algebraic Real Field
5 sage: W.tensor_module(2, 2)
6 Free module of type(2,2) tensors
7 on the 3dimensional vector space over the Algebraic Real Field
It is represented by an instance of the class FiniteRankFreeModule from sage.tensor.modules. These modules are the foundation for the multilinear algebra developed by the SageManifolds project.
Connecting FiniteRankFreeModule and free modules with distinguished basis
Given a basis of a FiniteRankFreeModule, the new method isomorphism_with_fixed_basis (#30094) constructs an isomorphism from the FiniteRankFreeModule to a free module in the category ModulesWithBasis. By default, it uses a CombinatorialFreeModule:
1 sage: V = FiniteRankFreeModule(QQ, 3, start_index=1); V
2 3dimensional vector space over the Rational Field
3 sage: basis = e = V.basis("e"); basis
4 Basis (e_1,e_2,e_3) on the 3dimensional vector space over the
5 Rational Field
6 sage: phi_e = V.isomorphism_with_fixed_basis(basis); phi_e
7 Generic morphism:
8 From: 3dimensional vector space over the Rational Field
9 To: Free module generated by {1, 2, 3} over Rational Field
10 sage: phi_e(e[1] + 2 * e[2])
11 e[1] + 2*e[2]
Eigenvalues and eigenvectors
Experimental functions for computing eigenvalues and eigenvectors in arbitrary precision (via Arb) including error bounds have been added. #30393
sage: from sage.matrix.benchmark import hilbert_matrix sage: mat = hilbert_matrix(3).change_ring(CBF) sage: mat.eigenvalues() [[1.40831892712365 +/ 7.16e15] + [+/ 2.02e15]*I, [0.12232706585391 +/ 6.49e15] + [+/ 2.02e15]*I, [0.00268734035577 +/ 5.60e15] + [+/ 2.02e15]*I]
Solving generalized eigenvalue problems Av = λBv for two square matrices A, B over RDF or CDF is now supported (via SciPy) #29243. A generalized eigenvalue λ is defined to be a root of the polynomial det(A  λ B) if this polynomial is not constantly zero.
sage: A = matrix.identity(RDF, 2) sage: B = matrix(RDF, [[3, 5], [6, 10]]) sage: D, V = A.eigenmatrix_right(B); D # tol 1e14 [0.07692307692307694 0.0] [ 0.0 +infinity] sage: λ = D[0, 0] sage: v = V[:, 0] sage: (A * v  B * v * λ).norm() < 1e14 True sage: A.eigenvalues(B, homogeneous=True) [(0.9999999999999999, 13.000000000000002), (0.9999999999999999, 0.0)]
Other improvements
Sage 9.2 has also merged a number of improvements to sage.tensor.modules: #30094, #30169, #30179, #30181, #30194, #30250, #30251, #30254, #30255, #30287
Polyhedral geometry
New features
It is now possible to choose which backend to use to compute regions of hyperplane arrangements 29506:
It is now possible to compute the slack matrix of a polyhedron 29838:
It is now possible to apply an affine transformation on a polyhedron 30327:
1 sage: M = random_matrix(QQ,3,3)
2 sage: v = vector(QQ,(1,2,3))
3 sage: F = AffineGroup(3, QQ)
4 sage: f = F(M, v); f
5 [ 0 0 2] [1]
6 x > [ 0 1 0] x + [2]
7 [ 1 1 1/2] [3]
8 sage: cube = polytopes.cube()
9 sage: f * cube
10 A 3dimensional polyhedron in QQ^3 defined as the convex hull of 8 vertices
11 sage: f(cube) # also works
12 A 3dimensional polyhedron in QQ^3 defined as the convex hull of 8 vertices
Implementation improvements
 It is now possible to set up polyhedra with both Vrep and Hrep in the following constructions:
The generation of regions of hyperplane arrangement has been improved 29661
Ehrhart related functions are now cached 29196
Obtaining incidence matrix and combinatorial polyhedron is much faster for integer and rational polyhedra 29837, 29841
The test coverage for the various backends for polyhedral computations has been improved by using _test_... methods to the abstract base class Polyhedron_base, in addition to doctests. See Metaticket #29842 Run a more stable test suite on polyhedra.
The face lattice can be obtained in reasonable time and no longer leaks memory 28982
There are also some bug fixes and other improvements. For more details see the release notes for optimization and polyhedral geometry software interactions in Sage.
Combinatorics
Reduction from Dancing links to SAT or MILP
It is now possible to solve an instance of an exact cover problem using a reduction from a dancing links instance to SAT 29338 or MILP 29955:
1 sage: from sage.combinat.matrices.dancing_links import dlx_solver
2 sage: rows = [[0,1,2], [3,4,5], [0,1], [2,3,4,5], [0], [1,2,3,4,5]]
3 sage: d = dlx_solver(rows)
4 sage: d.one_solution()
5 [1, 0]
6 sage: d.one_solution_using_sat_solver('cryptominisat')
7 [2, 3]
8 sage: d.one_solution_using_sat_solver('glucose')
9 [2, 3]
10 sage: d.one_solution_using_sat_solver('glucosesyrup')
11 [2, 3]
12 sage: d.one_solution_using_sat_solver('picosat')
13 [4, 5]
14 sage: d.one_solution_using_milp_solver()
15 [0, 1]
16 sage: d.one_solution_using_milp_solver('Gurobi')
17 [0, 1]
Polyomino tilings
It is now possible to find a surrounding of a polyomino with copies of itself, see 29160. This is based on the dancing links solver in Sage. This is motivated by the Heesch's problem. An example is below:
sage: from sage.combinat.tiling import Polyomino sage: H = Polyomino([(1, 1), (1, 4), (1, 7), (0, 0), (0, 1), (0, 2), ....: (0, 3), (0, 4), (0, 5), (0, 6), (0, 7), (0, 8), (1, 1), (1, 2), ....: (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (1, 8), (2, 0), (2, 2), ....: (2, 3), (2, 5), (2, 6), (2, 8)]) sage: H.show2d()
sage: %time solution = H.self_surrounding(10, ncpus=8) CPU times: user 1.69 s, sys: 1.08 s, total: 2.77 s Wall time: 3.85 s sage: G = sum([p.show2d() for p in solution], Graphics()) sage: G
Fully commutative elements of Coxeter groups
It is now possible by 30243 to enumerate and work with the fully commutative elements of a Coxeter group. Methods to compute *star operations* and plot the *heaps* of such elements are also included.
1 sage: A3 = CoxeterGroup(['A', 3])
2 sage: FCA3 = A3.fully_commutative_elements()
3 sage: FCA3.category()
4 Category of finite enumerated sets
5 sage: FCA3.list()
6 [[],
7 [1],
8 [2],
9 ...
10 [1, 3, 2],
11 [1, 2, 3],
12 [2, 1, 3, 2]]
13 sage: B8 = CoxeterGroup(['B', 8])
14 sage: FCB8 = B8.fully_commutative_elements()
15 sage: len(FCB8) # long time (7 seconds)
16 14299
17 sage: B6 = CoxeterGroup(['B', 6])
18 sage: FCB6 = B6.fully_commutative_elements()
19 sage: w = FCB6([1, 6, 2, 5, 4, 6, 5])
20 sage: w.coset_decomposition({5, 6})
21 ([6, 5, 6], [1, 2, 4, 5])
22 sage: w.star_operation({5,6}, 'lower')
23 [1, 5, 2, 4, 6, 5]
24 sage: FCB6([3, 2, 4, 3, 1]).plot_heap()
BIBDs with lambda>1
Sage can now construct a number of balanced incomplete block designs (BIBDs) with lambda>1, in particular, all the known biplanes (i.e. symmetric BIBDs with lambda=2).
Finite generalized polygons
Sage can now construct generalized quadrangles, hexagons, and octagons, and generalized quadrangles with a spread.
Graph theory
Distance regular graphs generators
A small database of constructions for distance regular graphs is now available. 15 sporadic distance regular graphs and 8 infinite families can now be constructed. Now Sage is capable of constructing all known families of distanceregular graphs with unbounded diameter.
Some code examples:
Diameter, radius, eccentricities
Stateoftheart algorithms for computing the diameter, the radius and the eccentricities of (directed) (weighted) graphs are now available 29657.
1 sage: G = graphs.RandomBarabasiAlbert(10000, 2)
2 sage: %time G.diameter(algorithm='DHV') # now default for undirected unweighted graphs
3 CPU times: user 74.4 ms, sys: 2.81 ms, total: 77.2 ms
4 Wall time: 75.8 ms
5 9
6 sage: %time G.diameter(algorithm='BFS')
7 CPU times: user 573 ms, sys: 4.04 ms, total: 577 ms
8 Wall time: 576 ms
9 9
More constructions
linear codes got a function to compute its coset graph; undirected graphs got a method to quickly compute their antipodal quotients.
More iterators
Some methods have been turned to iterators to avoid returning long lists 23002, 30470.
1 sage: g = graphs.PathGraph(5)
2 sage: bridges = g.bridges()
3 sage: next(bridges)
4 (3, 4, None)
5 sage: next(bridges)
6 (2, 3, None)
7 sage: next(bridges)
8 (1, 2, None)
9 sage: next(bridges)
10 (0, 1, None)
11 sage: next(bridges)
12 
13 StopIteration Traceback (most recent call last)
14 <ipythoninput38cf858e1a0a30> in <module>
15 > 1 next(bridges)
16
17 StopIteration:
Implementation improvements
Truly linear time implementation of lex_BFS 11736
 Faster and memory efficient implementation for computing the distances distribution, Wiener index and Szeged index. These new implementations have linear memory consumption rather than quadratic.
We started improving the backends to speed up basic operations (edge and vertex iterations, subgraph, etc.). More to come for next release 28895.
Commutative algebra
Laurent polynomials
Rings of Laurent polynomials now support ideal creation and manipulation 29512:
sage: L.<x,y,z> = LaurentPolynomialRing(QQ, 3) sage: I = L.ideal([(x+y+z)^3+x*y, x^2+y^2+z^2]) sage: I.groebner_basis() (y^4 + 4*x*y*z^2 + y^2*z^2 + 2*x*z^3 + 2*y*z^3  z^4 + 3/2*x*y*z + 1/4*x*z^2 + 1/4*y*z^2  1/4*z^3 + 1/8*x*y, x*y^2  y^3 + 3*x*y*z + x*z^2  z^3 + 1/2*x*y, x^2 + y^2 + z^2) sage: (x^3+y^3+z^3) in I False sage: x + x^1*y^2 + x^1*z^2 in I True
Motivic multiple zetas
The ring of motivic multiple zeta values has been implemented, using algorithms of Francis Brown. It allows to compute at least up to weight 12 22713.
sage: Multizeta(1,2)**2 12*ζ(1,1,1,3) + 6*ζ(1,1,2,2) + 2*ζ(1,2,1,2) sage: Multizeta(1,2)==Multizeta(3) True sage: Multizeta(2,3,4).n(100) 0.0029375850405618494701189454256
The numerical evaluation is based on PARI implementation.
Power series
There is a new method to compute the coefficients in the Jacobi continued fraction expansion of a power series 29789.
sage: t = QQ[['t']].0 sage: f = sum(factorial(n)*t**n for n in range(20)).O(20) sage: f.jacobi_continued_fraction() ((1, 1), (3, 4), (5, 9), (7, 16), (9, 25), (11, 36), (13, 49), (15, 64), (17, 81))
Ring homomorphisms
For many polynomial ring homomorphisms, the methods inverse, is_invertible, is_injective, is_surjective, kernel and inverse_image have been implemented. This covers not only polynomial rings, but also quotient rings, number fields and Galois fields. #9792 #29723
sage: R.<x,y,z> = QQ[] sage: sigma = R.hom([x  2*y*(z*x+y^2)  z*(z*x+y^2)^2, y + z*(z*x+y^2), z], R) sage: tau = sigma.inverse(); tau Ring endomorphism of Multivariate Polynomial Ring in x, y, z over Rational Field Defn: x > y^4*z  2*x*y^2*z^2  x^2*z^3 + 2*y^3 + 2*x*y*z + x y > y^2*z  x*z^2 + y z > z sage: (tau * sigma).is_identity() True
The method inverse_image can be used to test whether an element is contained in a subalgebra:
sage: R.<s,t> = PolynomialRing(QQ) sage: S.<x,y,z,w> = PolynomialRing(QQ) sage: f = S.hom([s^4, s^3*t, s*t^3, t^4], R) sage: f.inverse_image(R.ideal(0)) Ideal (y*z  x*w, z^3  y*w^2, x*z^2  y^2*w, y^3  x^2*z) of Multivariate Polynomial Ring in x, y, z, w over Rational Field sage: f.inverse_image(s^3*t^4*(s+t)) x*w + y*w sage: f.inverse_image(s^2*t^2) ... ValueError: element s^2*t^2 does not have preimage
Splitting Algebras
These are implemented recursively as quotient rings:
sage: from sage.algebras.splitting_algebra import SplittingAlgebra ....: L.<u, v, w> = LaurentPolynomialRing(ZZ); x = polygen(L) ....: S.<X, Y> = SplittingAlgebra(x^3  u*x^2 + v*x  w) sage: S.splitting_roots() [X, Y, Y  X + u] sage: ~X (w^1)*X^2 + (u*w^1)*X + v*w^1 sage: ~Y ((w^1)*X)*Y + (w^1)*X^2 + (u*w^1)*X
See also the reference manual.
Manifolds
diff function for exterior derivatives
It is now possible to invoke diff to compute the differential (exterior derivative) of a differentiable form (#29953). For instance, for a scalar field:
sage: M = Manifold(2, 'M') sage: X.<x,y> = M.chart() sage: f = M.scalar_field(x^2*y, name='f') sage: diff(f) 1form df on the 2dimensional differentiable manifold M sage: diff(f).display() df = 2*x*y dx + x^2 dy
and for a 1form:
sage: a = M.one_form(y, x, name='a'); a.display() a = y dx + x dy sage: diff(a) 2form da on the 2dimensional differentiable manifold M sage: diff(a).display() da = 2 dx/\dy
Dot and cross products of vector fields along a curve
The methods dot_product(), cross_product() and norm() can be now be used for vector fields defined along a differentiable map, the codomain of which is a Riemannian manifold (#30318). Previously, these methods worked only for vector fields on a Riemannian manifold, i.e. along the identity map. An important subcase is of course that of a curve in a Riemannian manifold. For instance, let us consider a helix C in the Euclidean space E^{3} parametrized by its arc length s:
sage: E.<x,y,z> = EuclideanSpace() sage: R.<s> = RealLine() sage: C = E.curve((2*cos(s/3), 2*sin(s/3), sqrt(5)*s/3), (s, oo, +oo), ....: name='C') sage: C.display() C: R > E^3 s > (x, y, z) = (2*cos(1/3*s), 2*sin(1/3*s), 1/3*sqrt(5)*s)
The tangent vector field T=C' has a unit norm since the parameter s is the arc length:
sage: T = C.tangent_vector_field() sage: T.display() C' = 2/3*sin(1/3*s) e_x + 2/3*cos(1/3*s) e_y + 1/3*sqrt(5) e_z sage: norm(T) Scalar field C' on the Real interval (0, 6*pi) sage: norm(T).expr() 1
We introduce the unit normal vector N via the derivative of T:
sage: T_prime = R.vector_field([diff(T[i], s) for i in E.irange()], dest_map=C, ....: name="T'") sage: N = T_prime / norm(T_prime) sage: N.display() cos(1/3*s) e_x  sin(1/3*s) e_y
and we get the binormal vector B as the cross product of T and N:
sage: B = T.cross_product(N) sage: B Vector field along the Real number line R with values on the Euclidean space E^3 sage: B.display() 1/3*sqrt(5)*sin(1/3*s) e_x  1/3*sqrt(5)*cos(1/3*s) e_y + 2/3 e_z
We can then form the FrenetSerret frame:
sage: FS = R.vector_frame(('T', 'N', 'B'), (T, N, B), ....: symbol_dual=('t', 'n', 'b')) sage: FS Vector frame (R, (T,N,B)) with values on the Euclidean space E^3
and check that it is orthonormal:
sage: matrix([[u.dot(v).expr() for v in FS] for u in FS]) [1 0 0] [0 1 0] [0 0 1]
The FrenetSerret formulas, expressing the curvature and torsion of C, are obtained as:
sage: N_prime = R.vector_field([diff(N[i], s) for i in E.irange()], ....: dest_map=C, name="N'") sage: B_prime = R.vector_field([diff(B[i], s) for i in E.irange()], ....: dest_map=C, name="B'") sage: for v in (T_prime, N_prime, B_prime): ....: v.display(FS) ....: T' = 2/9 N N' = 2/9 T + 1/9*sqrt(5) B B' = 1/9*sqrt(5) N
Orientability of manifolds and vector bundles
It is now possible to define an orientation on a differentiable manifold and on a vector bundle (#30178). Orientations of topological manifolds have also been introduced, according to this definition.
Euclidean spaces as metric spaces
Euclidean spaces have been endowed with a distance function and have been set in the category of complete metric spaces (#30062):
sage: E.<x,y> = EuclideanSpace() sage: p = E((1,0)) # the point of coordinates (1,0) sage: q = E((0,2)) # the point of coordinates (0,2) sage: d = E.dist # the distance function sage: d(p,q) sqrt(5) sage: p.dist(q) sqrt(5) sage: E.category() Join of Category of smooth manifolds over Real Field with 53 bits of precision and Category of complete metric spaces
Bundle connections
Bundle connections have been improved (#30208) and their action on vector fields and sections has been implemented (#30209).
Internal code improvements and bug fixes
Many improvements/refactoring of the code have been performed in this release:
In addition, various bugs have been fixed: #30108, #30112, #30191, #30275, #30289, #30320, #30401, #30519.
Algebra
Lie Conformal Algebras
Implemented Lie conformal algebras and superalgebras. Here are some examples of their usage:
sage: V = lie_conformal_algebras.Virasoro(QQ); V The Virasoro Lie conformal algebra over Rational Field sage: V.inject_variables() Defining L, C sage: L.bracket(L) {0: TL, 1: 2*L, 3: 1/2*C} sage: L.T(2).bracket(L) {2: 2*TL, 3: 12*L, 5: 10*C} sage: V = lie_conformal_algebras.NeveuSchwarz(QQ) sage: V.some_elements() [L, G, C, TG, TG + 4*T^(2)G, 4*T^(2)G] sage: W = lie_conformal_algebras.FreeFermions(QQbar, 2); W The free Fermions super Lie conformal algebra with generators (psi_0, psi_1, K) over Algebraic Field sage: W.inject_variables() Defining psi_0, psi_1, K sage: psi_0.bracket(psi_1.T()) {} sage: psi_0.bracket(psi_0.T()) {1: K} sage: psi_0.is_even_odd() 1
For documentation on implemented features see Lie Conformal Algebra. For a list of implemented examples see Lie Conformal Algebra Examples.
Differential Weyl algebra
The action of differential operators from the Weyl algebra on polynomials has been implemented #29928:
sage: W.<x,y> = DifferentialWeylAlgebra(QQ) sage: dx, dy = W.differentials() sage: dx.diff(x^3) 3*x^2 sage: (x*dx + dy + 1).diff(x^4*y^4 + 1) 5*x^4*y^4 + 4*x^4*y^3 + 1
Improved Unicode support
Unicode identifiers
Python 3 made much improved support for Unicode available, and Sage 9.2 has merged several Unicode improvements. Note that Python does not allow arbitrary Unicode characters in identifiers but only word constituents. So before you get excited about using emojis... note that they cannot be used:
However, we can use letters from various alphabets. The updated IPython allows us to type them using latex and unicode tab completion:
1 sage: μ, ν, ξ = 1, 2, 3 # type \mu<TAB>,
2 # \nu<TAB>, ...
3 sage: SR('λ + 2λ')
4 3*λ
5 sage: var('α', domain='real')
6 α
7 sage: Ш = EllipticCurve('389a').sha()
8 # type \CYR<TAB> CAP<TAB>
9 # LET<TAB> SHA<TAB><ENTER>
10 sage: Ш
11 TateShafarevich group for the Elliptic Curve
12 defined by y^2 + y = x^3 + x^2  2*x over Rational Field
13 sage: GelʹfandT͡setlinPattern = GelfandTsetlinPattern
14 # type \MODIFIER LETTER
15 # PRIME<TAB><ENTER>
16 # for the romanized soft mark
17 sage: ГельфандЦетлинPattern = GelʹfandT͡setlinPattern
18 sage: ГельфандЦетлинPattern([[3, 2, 1], [2, 1], [1]]).pp()
19 3 2 1
20 2 1
21 1
22 sage: 四次方(x) = x^4
23 sage: 四次方(3)
24 81
We can use math accents...
... and have fun with modifier letters:
1 sage: ℚ̄ = QQbar # type \bbQ<TAB>\bar<TAB>
2 sage: %display unicode_art
3 sage: A = matrix(ℚ̄, [[1, 2*I], [3*I, 4]]); A
4 ⎛ 1 2*I⎞
5 ⎝3*I 4⎠
6 sage: Aᵀ = A.transpose() # type A\^T<TAB><ENTER>
7 sage: Aᵀ
8 ⎛ 1 3*I⎞
9 ⎝2*I 4⎠
10 sage: Aᴴ = A.conjugate_transpose()
11 # type A\^H<TAB><ENTER>
12 sage: Aᴴ
13 ⎛ 1 3*I⎞
14 ⎝2*I 4⎠
15 sage: C = Cone([[1, 1], [0, 1]])
16 sage: Cᵒ = C.dual(); Cᵒ # type C\^o<TAB><ENTER>
17 2d cone in 2d lattice M
But note that Python normalizes identifiers, so the following variants are not distinguished:
1 sage: AT == Aᵀ, AH == Aᴴ, Co == Cᵒ
2 (True, True, True)
3 sage: ℚ = QQ # type \bbQ<TAB><ENTER>
4 sage: ℚ
5 Rational Field
6 sage: Q = 42
7 sage: ℚ
8 42
9 sage: F = 1
10 sage: 𝐹, 𝐅, 𝓕, 𝕱, 𝗙, 𝘍, 𝙁, 𝙵 # type \itF<TAB>, \bfF<TAB>,
11 # \scrF<TAB>, \frakF<TAB>,
12 # \sansF<TAB>, ...
13 (1, 1, 1, 1, 1, 1, 1, 1)
We have also added a few Unicode aliases for global constants and functions.
See Metaticket #30111: Unicode support for more information.
Unicode characters allowed in tensor index notation
Greek letters (and more generally any Unicode nondigit wordconstituent character) are now allowed in index notation for tensors (#29248). For instance, taking the trace of a type(1,1) tensor field:
sage: E.<x,y> = EuclideanSpace() sage: t = E.tensor_field(1, 1, [[x, 1], [0, y]]) sage: t['^μ_μ'] Scalar field on the Euclidean plane E^2 sage: t['^μ_μ'] == t.trace() True
Unicode art
#30119 Implemented a general function for writing integers as unicode sub/superscripts.
In #29205, some Lie algebra elements now have better unicode support:
sage: L = LieAlgebra(QQ, cartan_type="A2", representation='matrix') sage: unicode_art(L.an_element()) ⎛ 1 1 0⎞ ⎜ 1 0 1⎟ ⎝ 0 1 1⎠ sage: L = lie_algebras.Heisenberg(QQ, 2) sage: unicode_art(sum(L.basis())) p₁ + p₂ + q₁ + q₂ + z sage: L = lie_algebras.VirasoroAlgebra(QQ) sage: unicode_art(L.an_element()) d₋₁ + d₀  1/2 + c sage: L = LieAlgebra(QQ, cartan_type=['A',1,1]) sage: unicode_art(L.an_element()) ( alpha[1] + alphacheck[1] + alpha[1] )⊗t⁰ + ( alpha[1] )⊗t¹ + ( alpha[1] )⊗t⁻¹ + c + d sage: L.<x,y> = LieAlgebra(QQ) sage: unicode_art(x.bracket(y)) [x, y] sage: L = LieAlgebra(QQ, cartan_type=['A',2], representation="compact real") sage: unicode_art(L.an_element()) ⎛ i i + 1 i + 1⎞ ⎜i  1 i i + 1⎟ ⎝i  1 i  1 2*i⎠
As part of #29696, TemperleyLieb diagrams now have unicode (and ascii) art:
sage: from sage.combinat.diagram_algebras import TL_diagram_ascii_art sage: TL = [(15,12), (14,13), (11,15), (10,14), (9,6), ....: (8,7), (5,4), (3,1), (2,1), (2,3), (4,5), ....: (6,11), (7, 8), (9,10), (12,13)] sage: TL_diagram_ascii_art(TL, use_unicode=False) o o o o o o o o o o o o o o o  `` ``  `` ``  ``    ``    .`  `.  .`  ..   .. ..  ..  ..     ..  o o o o o o o o o o o o o o o sage: TL_diagram_ascii_art(TL, use_unicode=True) ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ │ ╰─╯ ╰─╯ │ ╰─╯ ╰─╯ │ ╰─╯ │ │ │ ╰─────────╯ │ │ │ ╭───────╯ │ ╰───╮ │ ╭───────╯ │ ╭─────╮ │ │ ╭─────╮ ╭─╮ │ ╭─╮ │ ╭─╮ │ │ │ │ ╭─╮ │ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬
Configuration and build changes
Initial configuration with ./configure required
Sage 9.1 introduced informational messages at the end of a ./configure run regarding system packages. To make sure that these messages are not overlooked, Sage 9.2 no longer invokes ./configure when you type make in an unconfigured source tree. See sagedevel: require "./configure" before "make", #29316.
Support for gcc/gfortran 10.x added
All standard Sage packages have been upgraded in Sage 9.2 so that they build correctly using gcc/gfortran 10.x. The Sage ./configure script therefore now accepts these compiler versions.
Selecting a system Python to use for Sage's venv
The configure script in Sage 9.2 has been changed so it only looks for a binary named python3 in your PATH. If Sage cannot find a suitable python3 in your PATH, it will build its own copy of Python 3.8.5. Sage no longer looks for versioned Python binaries such as python3.7, #30546.
To configure Sage to use a specific Python installation, you can use ./configure withpython=/path/to/python3. The configure script will test whether this installation is suitable for Sage and will exit with an error otherwise.
System package information for more distributions
System package information has been added for Gentoo Linux, FreeBSD, Void Linux, and NixOS.
System package information for optional packages at runtime
When a user tries to use a feature depending on an optional package is not installed, Sage now issues advice regarding the packages that should be installed to provide the feature  using either the system package manager, pip, or (in the Sage distribution) sage i #30606.
(For packagers: For this to work, either SAGE_ROOT/build/pkgs/*/distros/ and SAGE_ROOT/build/bin/{sagegetsystempackages, sageguesspackagesystem, sageprintsystempackagecommand} need to be installed, or alternative implementations of these scripts need to be provided.)
For developers: Changes to the build system of sagelib
Let's talk about setup.py. The build system of the Sage library, in build/pkgs/sagelib/src/setup.py, is based on distutils (not setuptools); it is implemented in the package sage_setup. In particular, it implements its own version of source code discovery methods similar to setuptools.find_packages: sage_setup.find.find_python_sources. Because of source discovery, developers can add new Python modules and packages under src/sage/ simply by creating files and directories; it is not necessary to edit setup.py.
Prior to Sage 9.2, the situation was different for Cython extensions. They had to be listed in src/module_list.py, either one by one, or using glob patterns such as * and **. Sage 9.2 has eliminated the need for src/module_list.py by extending sage_setup.find.find_python_sources; it now also finds Cython modules in the source tree (Trac #29701).
Some Cython modules need specific compiler and linker flags. Sage 9.2 has moved all of these flags from Extension options in src/module_list.py to distutils: directives in the individual .pyx source files, see #29706 and Cython documentation.
Sage 9.2 has also changed the mechanism for conditionalizing a Cython extension module on the presence of a Sage package. Consider the module sage.graphs.graph_decompositions.tdlib as an example. Prior to Sage 9.2, this module was declared as an OptionalExtension, conditional on the SPKG tdlib, in src/module_list.py. The new mechanism is as follows. src/setup.py maps the SPKG name tdlib to the "distribution name" sagetdlib. At the top of the Cython source file src/sage/graphs/graph_decompositions/tdlib.pyx, there is a new directive sage_setup: distribution = sagetdlib. Now the source discovery in sage_setup.find.find_python_sources includes this Cython module only if the SPKG tdlib is installed and current.
New development tools
Testing and linting with tox
tox is a popular package that is used by a large number of Python projects as the standard entry point for testing and linting.
Sage 9.1 started to use tox for portability testing of the Sage distribution, which requires an installation of tox in the system python.
Sage 9.2 has added a tox configuration (src/tox.ini) for the (more typical) use of tox for testing and linting of the Sage library #30453. This provides an entry point for various testing/linting methods that is more idiomatic from the viewpoint of the Python community.
The commands sage t, sage coverage, sage coverageall, and sage startuptime are repackaged as sage tox, as the following output from sage advanced indicates:
$ ./sage advanced SageMath version 9.2 ... Testing files: ... tox [options] <filesdirs>  general entry point for testing and linting of the Sage library e <envlist>  run specific test environments (default: run all) doctest  run the Sage doctester (same as "sage t") coverage  information about doctest coverage of files (same as "sage coverage[all]") startuptime  display how long each component of Sage takes to start up (same as "sage startuptime")
Three new linting methods are added:
pycodestyle  check against the Python style conventions of PEP8 relint  check whether some forbidden patterns appear (includes all patchbot patternexclusion plugins) codespell  check for misspelled words in source code
This functionality is available after installing the optional tox package using sage i tox (or having tox available in the system).
Reusable wheels for the Python packages built by the Sage distribution
Sage 9.2 has changed the build process of all Python packages in the Sage distribution so that wheels are built and stored in $SAGE_LOCAL/var/lib/sage/wheels/ #29500.
Users can install these wheels into virtual environments that use the same base python version, using standard tools such as pip install findlinks. Example:
$ local/bin/python3 m venv somevenv $ cd somevenv/ $ bin/pip3 install v noindex findlinks=../local/var/lib/sage/wheels/ Pillow
The installation is very fast because it does not involve compilation.
Note: Many of these wheels include extension modules that refer to libraries in the Sage installation in $SAGE_LOCAL with hardcoded paths. Therefore the wheels are not immediately suitable for distribution.
Cleaning
#29636: Delete changelog sections from all SPKG information files; they were deprecated in favor of using Trac years ago. The contributions of Sage developers maintaining SPKGs are documented by our historical changelogs.
Removing support for Python 2 allowed us to remove several backport packages in #29754
We also removed the deprecated SageNB (superseded a long time ago by the Jupyter notebook) in #29754 and several of its dependencies. For converting old Sage worksheet files (*.sws), the script sage sws2rst is available. (In Sage 9.0 and 9.1, it was available only in Python 2 builds of Sage; in #28838, it was ported to Python 3.)
Support for installing "oldstyle Sage packages" (.spkg files), deprecated in Sage 6.9, has been removed in #29289, after making the last two missing packages, cunningham_tables and polytopes_db_4d, available as normal optional Sage packages. Users who wish to package their own Sage code for distribution may find a list of external packages helpful, many of which follow best practices in packaging.
The use of sage.misc.package has been essentially eliminated from the Sage library by transitioning to sage.feature; see #30607, #30616.
Availability of Sage 9.2 and installation help
SageMath 9.2 was released on 20201024.
Sources
The Sage source code is available in the sage git repository, and the selfcontained source tarballs are available for download.
Sage 9.2 has been tested to compile from source on a wide variety of platforms, including:
Linux 64bit (x86_64)
 ubuntu{trusty,xenial,bionic,eoan,focal,groovy},
 debian{jessie,stretch,buster,bullseye,sid},
 linuxmint{17,18,19,19.3},
 fedora{26,27,28,29,30,31,32,33},
 centos{7,8},
 archlinux,
 slackware14.2,
 condaforge
Linux 32bit (i386)
 debianbuster
 ubuntueoan
macOS
 macOS Catalina and older (macOS 10.x, with Xcode 11.x or Xcode 12)
 optionally, using Homebrew
 optionally, using condaforge
note that building Sage from source on macOS 11 ("Big Sur") is not supported  use SageMath 9.3 instead
Windows (Cygwin64).
Binaries
Binaries for macOS. Do not use the App version of the binary macOS package (.app.dmg); it has been found to be problematic to install.
A signed and notarized macOS app which runs Sage 9.2 (new  Mar 2021)
Instructions for installing the macOS binary on Apple Silicon (M1)
If you get the error /usr/bin/env: 'python': No such file or directory. Error running the script 'relocateonce.py'.: Make sure that you have python installed in your system. On some recent Linux distributions, there is a python3 binary but no python binary. In this case, edit the first line of relocateonce.py to change python to python3.
Availability in distributions
The easiest way to install Sage 9.2 is through a distribution that provides it, see repology.org: sagemath.
On macOS, the easiest way to install Sage 9.2 is using condaforge
Installation FAQ
See README.md in the source distribution for installation instructions.
See sagerelease, sagedevel.