Implement as in multivariate polynomial ring
{{{
sage: L = LaurentPolynomialRing(QQ, 'x', 3)
sage: L.monomial(5, -7, 3)
x0^5*x1^-7*x2^3
}}}

URL: https://trac.sagemath.org/23727
Reported by: vdelecroix
Ticket author(s): Vincent Delecroix
Reviewer(s): Frédéric Chapoton

{{{
sage: l = libgap([0,1])
sage: l
sage: copy(l)
NULL
}}}

URL: https://trac.sagemath.org/23721
Reported by: vdelecroix
Ticket author(s): Vincent Delecroix
Reviewer(s): Aly Deines

Even the latest upstream version (3.3.4, not in Sage yet) still ships
with old autotools.

URL: https://trac.sagemath.org/23710
Reported by: jdemeyer
Ticket author(s): Jeroen Demeyer
Reviewer(s): François Bissey

as another step to py3

needs that a bot checks that nothing got broken

URL: https://trac.sagemath.org/23709
Reported by: chapoton
Ticket author(s): Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

as another step to py3

URL: https://trac.sagemath.org/23708
Reported by: chapoton
Ticket author(s): Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

improve getitem so that the following works
{{{
sage: l = libgap.eval('[[0,1],[2,3]]')
sage: l[0,0]
}}}
and implement setitem in order to be able to do
{{{
sage: l = libgap.eval('[[0,1],[2,3]]')
sage: l[0,0] = -3
}}}

URL: https://trac.sagemath.org/23704
Reported by: vdelecroix
Ticket author(s): Vincent Delecroix
Reviewer(s): Travis Scrimshaw

as another step to py3

URL: https://trac.sagemath.org/23698
Reported by: chapoton
Ticket author(s): Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

some occurrences in the doc, plus one in code of multi-polynomials

URL: https://trac.sagemath.org/23697
Reported by: chapoton
Ticket author(s): Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

Arithmetic with complex `I` may involve symbolic equation solving using
Maxima because `QQbar` calls `minpoly.roots` every time, which uses
Maxima's solve for symbolic output. Special code in
`sage/rings/polynomial/polynomial_element.pyx` reduces time 100x for the
first call (Maxima startup) and 2x for any following.

URL: https://trac.sagemath.org/23695
Reported by: rws
Ticket author(s): Ralf Stephan
Reviewer(s): Travis Scrimshaw

Free (sub)modules have a `dimension()` and a `degree()` which is the
dimension of the ambient space. To complement this, there should be
`codimension()` which is the difference between these two.

URL: https://trac.sagemath.org/23690
Reported by: jdemeyer
Ticket author(s): Jeroen Demeyer
Reviewer(s): Travis Scrimshaw

There's a typo in the `greedy` function; the following should be the
same.

{{{
sage: S = ClusterSeed(Matrix([[0,1],[-4,0]])); S
A seed for a cluster algebra of rank 2
sage: S.greedy(1,2)
(x1^4 + x0^2 + 2*x0 + 1)/(x0*x1^2)
sage: S.greedy(1,2,'by_combinatorics')
(x0*x1^4 + x1^4 + x0^2 + 2*x0 + 1)/(x0*x1^2)
}}}

URL: https://trac.sagemath.org/23688
Reported by: llpku
Ticket author(s): Li Li
Reviewer(s): Gregg Musiker

Followup to #23193.

Function field doctests take 10 minutes on reasonable machines. That's
way too much. The changes introduced here drop this again to 7/70
seconds for the doctests/long doctests.

URL: https://trac.sagemath.org/23683
Reported by: saraedum
Ticket author(s): Julian Rüth
Reviewer(s): Travis Scrimshaw, Frédéric Chapoton

as another step to py3

URL: https://trac.sagemath.org/23682
Reported by: chapoton
Ticket author(s): Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

Make the following two examples work
{{{
sage: G = cartesian_product([Zmod(3), Zmod(6), Zmod(5)])
sage: G((1,1,1)).additive_order()
30
}}}
and
{{{
sage: G1 = SymmetricGroup(3)
sage: G2 = SL(2,3)
sage: G = cartesian_product([G1,G2])
sage: G((G1.gen(0), G2.gen(1))).multiplicative_order()
12
}}}

URL: https://trac.sagemath.org/23679
Reported by: vdelecroix
Ticket author(s): Vincent Delecroix
Reviewer(s): Travis Scrimshaw

It is currently not doctested and broken.

URL: https://trac.sagemath.org/23677
Reported by: chapoton
Ticket author(s): Frédéric Chapoton
Reviewer(s): Marcelo Forets

step to python3

URL: https://trac.sagemath.org/23676
Reported by: chapoton
Ticket author(s): Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

and removing one scaring "import sage.all"

URL: https://trac.sagemath.org/23675
Reported by: chapoton
Ticket author(s): Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

As discussed in https://groups.google.com/forum/#!topic/sage-
devel/b1pWen3lrKA, the documentation of Fourier series as implemented in
piecewise-defined functions needs some improvement. This ticket provides
it, as well as a simplification of the user interface: the half-period
is now an optional argument; if not provided, it is inferred from the
domain of the piecewise-defined function:
{{{
sage: f = piecewise([((0, 2*pi), cos(x))])
sage: f.fourier_series_cosine_coefficient(1)  # results in TypeError in
Sage 8.0
1
sage: f.fourier_series_cosine_coefficient(1, pi)  # compatible with Sage
8.0
1
}}}
The ticket also corrects two bugs:
- computation of Fourier coefficients when the domain width does not
coincide with the period:
{{{
sage: f = piecewise([((0, 4*pi), cos(x))])
sage: f.fourier_series_cosine_coefficient(1, pi)  # yields 2 in Sage 8.0
1
}}}
- despite what it claimed, the method `fourier_series_partial_sum` did
not return ''S,,N,,(x)'' but ''S,,N-1,,(x)'' (this is because the
summation was governed by `srange(1, N)` instead of `srange(1, N+1)`);
we have now
{{{
sage: f = piecewise([((0, 2*pi), cos(x))])
sage: f.fourier_series_partial_sum(1, pi)  # yields 0 in Sage 8.0
cos(x)
}}}

URL: https://trac.sagemath.org/23672
Reported by: egourgoulhon
Ticket author(s): Eric Gourgoulhon
Reviewer(s): Richard L Lozes

In particular, the following example used to fail:
{{{
sage: P0.<x> = ZZ[]
sage: P1.<y> = Frac(P0)[]
sage: frac = (x/(x^2 + 1))*y + 1/(x^3 + 1)
sage: Frac(ZZ['x,y'])(frac)
(x^4*y + x^2 + x*y + 1)/(x^5 + x^3 + x^2 + 1)
}}}

URL: https://trac.sagemath.org/23664
Reported by: mmezzarobba
Ticket author(s): Marc Mezzarobba
Reviewer(s): Vincent Delecroix

The implementation contains an infinite loop that is broken when a
quantity is less than or equal to 1, however the loop never ends on the
following input:

{{{
sage: g=graphs.PetersenGraph()
sage: g.fractional_chromatic_index(solver="GLPK")
}}}

The problem seems to depend on the LP solver, using PPL seems to work
just fine. Issue seems to be GLPK and CBC/Coin.

Relevant sage-devel thread at [1].

Relevant ask.sagemath thread at [2].

[1]: https://groups.google.com/forum/#!topic/sage-devel/hsKxANYAeQI

[2]: https://ask.sagemath.org/question/38543/fractional_chromatic_index-
in-sage-76-gets-stuck-in-loop-adding-constraints/

URL: https://trac.sagemath.org/23658
Reported by: fidelbarrera
Ticket author(s): David Coudert
Reviewer(s): Dima Pasechnik

namely, do not use an element and its inverse as generators

This helps to fix partially #15341

URL: https://trac.sagemath.org/23656
Reported by: chapoton
Ticket author(s): Frédéric Chapoton
Reviewer(s): Vincent Delecroix

Some more easy cases as part of #10483.

URL: https://trac.sagemath.org/23638
Reported by: jdemeyer
Ticket author(s): Simon King, Ralf Stephan
Reviewer(s): Jeroen Demeyer, Ralf Stephan

1) `Posets.DiamondPoset(5).is_join_distributive(certificate=True)` gives
"global name 'DiGraph' is not defined".

2) Same applies to `is_meet_distributive`.

3) In the doc of `is_join_pseudocomplemented` a "greatest" should be
"least".

4) `is_vertically_decomposable` has some mutually exclusive properties.

5) Some seealso-blocks completed.

URL: https://trac.sagemath.org/23625
Reported by: jmantysalo
Ticket author(s): Jori Mäntysalo
Reviewer(s): Travis Scrimshaw

This ticket implements the fast method `is_trivial_zero` in coordinate
functions (classes `sage.manifolds.coord_func.CoordFunction` and
`sage.manifolds.coord_func_symb.CoordFunctionSymb`) and scalar fields
(class `sage.manifolds.scalarfield.ScalarField`). The method
`is_trivial_zero` is then used instead of `== 0` in tensor calculus,
especially in tensor assignement, where the check for zero is performed
because only nonzero components of a tensor are stored. This avoids the
call to Maxima, which can be problematic in some cases, as reported in
this [https://groups.google.com/forum/#!topic/sage-devel/teLBaNipBDM
sage-devel post]. Moreover, this results in a significant speed-up
factor. For instance, the computation time of the coefficients of the
Levi-Civita connection of the Kerr-Newman metric, as performed in cell
20 of this [http://nbviewer.jupyter.org/github/sagemanifolds/SageManifol
ds/blob/master/Worksheets/v1.0/SM_Kerr_Newman.ipynb example notebook],
is reduced from 5.37 s to  4.11 s (this is on a Intel Core i7-6700HQ
computer).

URL: https://trac.sagemath.org/23623
Reported by: egourgoulhon
Ticket author(s): Eric Gourgoulhon
Reviewer(s): Richard L Lozes

Routine upgrade. I have enough of seeing various doctests failing in
sage-on-gentoo when using this version.

tarball: [https://pypi.python.org/packages/52/67/d9ef9b5058d4a9e3f0ae641
ec151790622cbeb37f157de5773358e2bf3da/scipy-0.19.1.tar.gz]

URL: https://trac.sagemath.org/23594
Reported by: fbissey
Ticket author(s): François Bissey
Reviewer(s): Vincent Delecroix, Jean-Pierre Flori

Currently Gauss sums are available as method of Dirichlet Groups.

Here is code for an implementation for general finite fields.

added in src/sage/arith/misc.py

URL: https://trac.sagemath.org/23585
Reported by: chapoton
Ticket author(s): Frédéric Chapoton
Reviewer(s): Vincent Delecroix

URL: https://trac.sagemath.org/23560
Reported by: aschilling
Ticket author(s): Franco Saliola, Travis Scrimshaw, Anne Schilling
Reviewer(s): Ben Brubaker

Standard autotools packages allow variables such as CC to be set at
configure time using
{{{
./configure CC=/path/to/gcc
}}}
In this ticket, we add support for this. (The syntax already works, but
the settings right now only end up in the unused file `build/make
/Makefile-auto`.)

New configure tarball:
[https://trac.sagemath.org/raw-
attachment/ticket/22646/configure-235.tar.gz]

URL: https://trac.sagemath.org/22646
Reported by: mkoeppe
Ticket author(s): François Bissey
Reviewer(s): Matthias Koeppe, Jeroen Demeyer, Vincent Delecroix, John
Palmieri, Dima Pasechnik

Trac #21526: package autotools: '$SAGE_LOCAL/automake-1.11.6/share/aclocal': No such file or directory

I get this when installing `ecm`.

A workaround is:
{{{
cd automake-1.11.6/share/ && ln -s aclocal-1.11/ aclocal

}}}

Unclear if this a bug of a specific automake version, or a general
shortcoming of the installation scheme of our autotools. This needs
investigating.

URL: https://trac.sagemath.org/21526
Reported by: mkoeppe
Ticket author(s): Erik Bray
Reviewer(s): Matthias Koeppe

{{{
sage: QQx.<x> = QQ[[]]
sage: (1+x+O(x^100)).pade(2,2)
------------------------------------------------------------------------
---
IndexError                                Traceback (most recent call
last)
<ipython-input-2-9cb1c03a10fc> in <module>()
----> 1 (Integer(1)+x+O(x**Integer(100))).pade(Integer(2),Integer(2))

/Applications/sage-7.3/src/sage/rings/power_series_poly.pyx in
sage.rings.power_series_poly.PowerSeries_poly.pade (/Applications/sage-7
.3/src/build/cythonized/sage/rings/power_series_poly.c:12104)()
   1120         for i in range(1, n + 1):
   1121             for j in range(n + 1):
-> 1122                 mat[i, j] = c[m + i - j]
   1123         for j in range(n + 1):
   1124             mat[0, j] = z ** j

IndexError: list index out of range
}}}

Here is a quick fix:

{{{
for i in range(1, n + 1):
            for j in range(n + 1):
                if m + i - j < len(c): # new line
                # or
                # if m + i - j < self.degree():
                    mat[i, j] = c[m + i - j]
}}}

URL: https://trac.sagemath.org/21212
Reported by: edgarcosta
Ticket author(s): Edgar Costa
Reviewer(s): Frédéric Chapoton, Aly Deines, Travis Scrimshaw

The purpose of this patch is to improve the complexity of is_symmetric
function for words from quadratic to linear. To do this, we modify the
lps (longest palindrome suffix) function to make it linear (using the
last value of lps_lengths), and use a property of the longest palindrome
suffix of a square.

Apart of the interest for the function in itself, the goal is to make an
is_christoffel function that computes the answer in linear time (see
ticket #19159).

URL: https://trac.sagemath.org/19155
Reported by: nadialafreniere
Ticket author(s): Nadia Lafrenière
Reviewer(s): Mélodie Lapointe

Let ''f'' be a newform of weight ''k'' for Γ,,1,,(''n''), and let ''d''
be a divisor of ''n'' that is coprime to ''n/d''.  Then the Atkin-Lehner
operator ''W,,d,,'' acts by a ''pseudo-eigenvalue'' on ''f'': this is
(in the normalisation used by Sage) a complex number ''η'' of absolute
value ''d''^''k''/2 - 1^ such that ''W,,d,,f'' = ''ηf''^*^, where
''f''^*^ is a certain twist of ''f''.

The main goal of this ticket is to add a method
`Newform.atkin_lehner_action(d)` that, given a newform `f`, returns a
pair `(eta, g)`, where `eta` is the pseudo-eigenvalue of ''W,,d,,'' on
`f` and `g` is the appropriate twist of `f`.

This also fixes the following bug.  Consider the following newform of
weight 2 for Γ,,1,,(30) with coefficients in '''Q'''(''i''):
{{{
sage: f = Newforms(Gamma1(30), 2, names='a')[1]; f
q + a1*q^2 - a1*q^3 - q^4 + (a1 - 2)*q^5 + O(q^6)
sage: f.base_ring()
Number Field in a1 with defining polynomial x^2 + 1
sage: f.character()
Dirichlet character modulo 30 of conductor 5 mapping 11 |--> 1, 7 |-->
-1
}}}
The method `atkin_lehner_eigenvalue()` returns a nonsensical result:
{{{
sage: f.atkin_lehner_eigenvalue()
-2
}}}
This is just the upper left entry of the matrix of the Atkin-Lehner
operator ''W'',,30,, with respect to some basis of the space of modular
symbols attached to ''f'':
{{{
sage: f.modular_symbols(sign=1).atkin_lehner_operator()
Hecke module morphism Atkin-Lehner operator W_30 defined by the matrix
[-2 -3]
[ 1  2]
Domain: Modular Symbols subspace of dimension 2 of Modular Symbols space
...
Codomain: Modular Symbols subspace of dimension 2 of Modular Symbols
space ...
}}}

URL: https://trac.sagemath.org/18061
Reported by: pbruin
Ticket author(s): Peter Bruin, David Loeffler
Reviewer(s): Peter Bruin, David Loeffler, Frédéric Chapoton

In `sage/env.py`, the function `_add_variable_or_fallback` does some
variable substitution: if a string contains `$VAR`, it substitutes the
value of `VAR` in the dictionary `SAGE_ENV`. It does this by iterating
through the items of `SAGE_ENV`, which means that if both `VAR` and
`VAR_OTHER` are in `SAGE_ENV`, it's sort of a coin toss as to what
happens when it comes across `$VAR_OTHER`.

(This arose when I was working on #21469: the string `$SAGE_SRC_ROOT`
was being replaced by the value of `SAGE_SRC`, followed by the string
`_ROOT`.)

URL: https://trac.sagemath.org/23758
Reported by: jhpalmieri
Ticket author(s): John H. Palmieri
Reviewer(s): Matthias Koeppe

{{{
sage: X = SchubertPolynomialRing(ZZ)
sage: a = X([3,2,4,1])
sage: a.divided_difference(2)
python: : Unknown error -315708632
}}}
The error number is random every time. The result should be 0.

URL: https://trac.sagemath.org/23403
Reported by: tscrim
Ticket author(s): Travis Scrimshaw
Reviewer(s): Darij Grinberg

#23485 mistakenly used `def _richcmp_` instead of `cpdef _richmp_`.

URL: https://trac.sagemath.org/23750
Reported by: jdemeyer
Ticket author(s): Jeroen Demeyer
Reviewer(s): Travis Scrimshaw

Stopgap for #17696.

URL: https://trac.sagemath.org/23644
Reported by: saraedum
Ticket author(s): Julian Rüth, Maarten Derickx
Reviewer(s): David Roe

At #20913 pip was patched so that it works without ssl support, and this
patch was also submitted upstream. Upstream has since merged the patch
at ​https://github.com/pypa/pip/issues/1165 into 9.0.1. So it makes
sense to upgrade pip to this new version in order to have an unpatched
pip in sage.

The pip tarbal can be found at [https://pypi.python.org/packages/11/b6/a
bcb525026a4be042b486df43905d6893fb04f05aac21c32c638e939e447/pip-9.0.1.ta
r.gz#md5=35f01da33009719497f01a4ba69d63c9]

URL: https://trac.sagemath.org/23615
Reported by: mderickx
Ticket author(s): Maarten Derickx
Reviewer(s): Jeroen Demeyer

I have implemented rational reconstruct over a generic univariate
polynomial ring.

I'm not sure if I put in the right place, if not, please let me know.

URL: https://trac.sagemath.org/23534
Reported by: edgarcosta
Ticket author(s): Edgar Costa
Reviewer(s): Aly Deines, Travis Scrimshaw