sagemath · darijgr · Mar 14, 2025 · Mar 14, 2025
diff --git a/src/sage/matrix/matrix2.pyx b/src/sage/matrix/matrix2.pyx
@@ -3035,6 +3035,61 @@ cdef class Matrix(Matrix1):
         self.cache('minpoly', mp)
         return mp
 
+    def minpoly_lin(self, var='x', **kwds):
+        r"""
+        Return the minimal polynomial of ``self``.
+
+        This uses a purely linear-algebraic algorithm (essentially
+        Gaussian elimination, applied to the powers of ``self``).
+        It is slow but it requires no factorization of polynomials.
+
+        EXAMPLES::
+
+            sage: # needs sage.rings.finite_rings
+            sage: A = matrix(GF(9, 'c'), 4, [1,1,0,0, 0,1,0,0, 0,0,5,0, 0,0,0,5])
+            sage: A.minpoly_lin()
+            x^3 + 2*x^2 + 2*x + 1
+            sage: A.minpoly_lin()(A) == 0
+            True
+            sage: CF = CyclotomicField()
+            sage: i = CF.gen(4)
+            sage: A = matrix(CF, 4, [1,1,0,0, 0,1,0,0, 0,0,1+i,0, 0,0,0,-i])
+            sage: A.minpoly_lin()
+            x^4 - 3*x^3 + (4 - E(4))*x^2 + (-3 + 2*E(4))*x + 1 - E(4)
+
+        The default variable name is `x`, but you can specify
+        another name::
+
+            sage: # needs sage.rings.finite_rings
+            sage: A.minpoly_lin('y')
+            y^4 - 3*y^3 + (4 - E(4))*y^2 + (-3 + 2*E(4))*y + 1 - E(4)
+        """
+        f = self.fetch('minpoly')
+        if f is not None:
+            return f.change_variable_name(var)
+
+        n = self.nrows()
+        F = self.base_ring()
+        from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
+        P = PolynomialRing(F, "x")
+        x = P.gen()
+        from sage.modules.free_module_element import vector
+        from sage.matrix.constructor import matrix
+
+        pow = self**0
+        pows = []
+        for k in range(n+1):
+            pows.append(vector(pow.list()))
+            pow *= self
+            Mpows = matrix(pows)
+            try:
+                cs = Mpows.solve_left(vector(pow.list()))
+            except ValueError:
+                continue
+            mp = x**(k+1) - P.sum(cs[i] * x**i for i in range(k+1))
+            self.cache('minpoly', mp)
+            return mp
+
     def _test_minpoly(self, **options):
         """
         Check that :meth:`minpoly` works.

diff --git a/src/doc/en/thematic_tutorials/vector_calculus/vector_calc_change.rst b/src/doc/en/thematic_tutorials/vector_calculus/vector_calc_change.rst
 These formulas are automatically used if we ask to plot the grid of spherical
 coordinates in terms of Cartesian coordinates::

    sage: spherical.plot(cartesian, color={r:'red', th:'green', ph:'orange'})
    Graphics3d Object

 .. PLOT::
diff --git a/src/doc/en/thematic_tutorials/vector_calculus/vector_calc_curvilinear.rst b/src/doc/en/thematic_tutorials/vector_calculus/vector_calc_curvilinear.rst

 The Laplacian of a vector field::

    sage: Du = laplacian(u)
    sage: Du.display()
    Delta(u) = ((r^2*d^2(u_r)/dr^2 + 2*r*d(u_r)/dr - 2*u_r(r, th, ph)
     + d^2(u_r)/dth^2 - 2*d(u_theta)/dth)*sin(th)^2 - ((2*u_theta(r, th, ph)
diff --git a/src/sage/combinat/symmetric_group_algebra.py b/src/sage/combinat/symmetric_group_algebra.py
        and also projective modules)::

            sage: SGA = SymmetricGroupAlgebra(QQ, 5)
            sage: for la in Partitions(SGA.n):
            ....:     idem = SGA.ladder_idemponent(la)
            ....:     assert idem^2 == idem
            ....:     print(la, SGA.principal_ideal(idem).dimension())
diff --git a/src/sage/combinat/tiling.py b/src/sage/combinat/tiling.py

        ::

            sage: solution = H.self_surrounding(8, remove_incomplete_copies=False)
            sage: G = sum([p.show2d() for p in solution], Graphics())                   # needs sage.plot
        """
        # Define the box to tile
diff --git a/src/sage/doctest/forker.py b/src/sage/doctest/forker.py
        sage: W.start()
        sage: DC = DocTestController(DD, filename)
        sage: reporter = DocTestReporter(DC)
        sage: W.join()  # Wait for worker to finish
        sage: result = W.result_queue.get()
        sage: reporter.report(FDS, False, W.exitcode, result, "")
            [... tests, ...s wall]
            ....: except OSError:
            ....:     print("Write end of pipe successfully closed")
            Write end of pipe successfully closed
            sage: W.join()  # Wait for worker to finish
        """
        super().start()

            sage: W.start()
            sage: while W.rmessages is not None:
            ....:     W.read_messages()
            sage: W.join()
            sage: len(W.messages) > 0
            True
        """
            sage: FDS = FileDocTestSource(filename, DD)
            sage: W = DocTestWorker(FDS, DD)
            sage: W.start()
            sage: W.join()
            sage: W.save_result_output()
            sage: sorted(W.result[1].keys())
            ['cputime', 'err', 'failures', 'optionals', 'tests', 'walltime', 'walltime_skips']
diff --git a/src/sage/dynamics/arithmetic_dynamics/projective_ds.py b/src/sage/dynamics/arithmetic_dynamics/projective_ds.py
            sage: P.<x,y>=ProjectiveSpace(QQbar, 1)
            sage: E=EllipticCurve([1, 2])
            sage: f=P.Lattes_map(E, 2)
            sage: f.Lattes_to_curve(check_lattes=true)
            Elliptic Curve defined by y^2 = x^3 + x + 2 over Rational Field

        """
diff --git a/src/sage/interacts/test_jupyter.rst b/src/sage/interacts/test_jupyter.rst
    Exact value of the integral \(\displaystyle\int_{0}^{2}x^{2} +
    1\,\mathrm{d}x=4.666666666666668\)

    sage: test(interacts.calculus.function_tool)
    ...Interactive function <function function_tool at ...> with 7 widgets
      f: EvalText(value='sin(x)', description='f')
      g: EvalText(value='cos(x)', description='g')
diff --git a/src/sage/combinat/designs/orthogonal_arrays_find_recursive.pyx b/src/sage/combinat/designs/orthogonal_arrays_find_recursive.pyx
        sage: from sage.combinat.designs.orthogonal_arrays_find_recursive import find_recursive_construction
        sage: from sage.combinat.designs.orthogonal_arrays import is_orthogonal_array
        sage: count = 0
        sage: for n in range(10,150):
        ....:     k = designs.orthogonal_arrays.largest_available_k(n)
        ....:     if find_recursive_construction(k,n):
        ....:         count = count + 1
diff --git a/src/sage/algebras/fusion_rings/fusion_ring.py b/src/sage/algebras/fusion_rings/fusion_ring.py
            True
            sage: F41 = FusionRing("F4", 1)
            sage: fmats = F41.get_fmatrix()
            sage: fmats.find_orthogonal_solution(verbose=False)
            sage: b = F41.basis()
            sage: all(F41.s_ijconj(x, y) == F41._basecoer(F41.s_ij(x, y, base_coercion=False).conjugate()) for x in b for y in b)
            True
diff --git a/src/sage/functions/orthogonal_polys.py b/src/sage/functions/orthogonal_polys.py
        sage: x = SR.var('x')
        sage: n = ZZ.random_element(5, 5001)
        sage: a = QQ.random_element().abs() + 5
        sage: s = (  (n + 1)*ultraspherical(n + 1, a, x)
        ....:      - 2*x*(n + a)*ultraspherical(n, a, x)
        ....:      + (n + 2*a - 1)*ultraspherical(n - 1, a, x) )
        sage: s.expand().is_zero()
diff --git a/src/sage/rings/polynomial/multi_polynomial_libsingular.pyx b/src/sage/rings/polynomial/multi_polynomial_libsingular.pyx
            ....: # when Singular improves the algorithm or hardware gets faster, increase n.
            sage: f = z^n-2
            sage: g = z^2-z-x^2*y-x*y^3
            sage: alarm(0.5); f.quo_rem(g)
            Traceback (most recent call last):
            ...
            AlarmInterrupt
diff --git a/src/sage/graphs/base/static_sparse_graph.pyx b/src/sage/graphs/base/static_sparse_graph.pyx
        sage: while not G.is_strongly_connected():
        ....:     shuffle(r)
        ....:     G.add_edges(enumerate(r), loops=False)
        sage: spectral_radius(G, 1e-10)  # random  # long time
        (1.9997956006500042, 1.9998043797692782)

    The algorithm takes care of multiple edges::
diff --git a/src/sage/graphs/generators/distance_regular.pyx b/src/sage/graphs/generators/distance_regular.pyx
    EXAMPLES::

        sage: # long time, needs sage.modules sage.rings.finite_rings
        sage: G = graphs.shortened_000_111_extended_binary_Golay_code_graph()   # 25 s
        sage: G.is_distance_regular(True)
        ([21, 20, 16, 9, 2, 1, None], [None, 1, 2, 3, 16, 20, 21])

    TESTS::

        sage: # needs sage.modules sage.rings.finite_rings
        sage: G = graphs.GrassmannGraph(2, 6, 3)        # long time
        sage: G.is_distance_regular(True)       # long time
        ([98, 72, 32, None], [None, 1, 9, 49])
        sage: G = graphs.GrassmannGraph(3, 4, 2)
         62
         sage: G.is_distance_regular(True)
         ([6, 5, 5, None], [None, 1, 1, 6])
         sage: G = graphs.DoubleGrassmannGraph(3, 2)    # long time                     # needs sage.rings.finite_rings
         sage: G.order()                        # long time                             # needs sage.rings.finite_rings
         2420
         sage: G.is_distance_regular(True)      # long time                             # needs sage.rings.finite_rings
        Half 4 Cube: Graph on 8 vertices
        sage: graph_with_classical_parameters(3, 2, 0, 2, 9)                            # needs sage.libs.gap
        Symplectic Dual Polar Graph DSp(6, 2): Graph on 135 vertices
        sage: graph_with_classical_parameters(3, 2, 2, 14, 7)   # long time             # needs sage.symbolic
        Grassmann graph J_2(6, 3): Graph on 1395 vertices
        sage: graph_with_classical_parameters(3, -2, -2, 6, 6)  # optional - gap_package_atlasrep internet
        Generalised hexagon of order (2, 8): Graph on 819 vertices
diff --git a/src/sage/groups/fqf_orthogonal.py b/src/sage/groups/fqf_orthogonal.py

    TESTS::

        sage: for p in primes_first_n(7)[1:]: # long time
        ....:     q = matrix.diagonal(QQ, 3 * [2/p])
        ....:     q = TorsionQuadraticForm(q)
        ....:     assert q.orthogonal_group().order()==GO(3, p).order()
diff --git a/src/sage/groups/finitely_presented_named.py b/src/sage/groups/finitely_presented_named.py
        (True, 1)
        sage: A3 = groups.presentation.Alternating(3); A3.order(), A3.as_permutation_group().is_cyclic()
        (3, True)
        sage: A8 = groups.presentation.Alternating(8); A8.order()
        20160
    """
    from sage.groups.perm_gps.permgroup_named import AlternatingGroup
diff --git a/src/sage/groups/perm_gps/permgroup_named.py b/src/sage/groups/perm_gps/permgroup_named.py
        [ 2  0 -1 -2  0  1]
        [ 2  0 -1  2  0 -1]
        sage: def numgps(n): return ZZ(libgap.NumberSmallGroups(n))
        sage: all(SmallPermutationGroup(n,k).id() == [n,k]
        ....:     for n in [1..64] for k in [1..numgps(n)])  # long time (180s)
        True
        sage: H = SmallPermutationGroup(6,1)
diff --git a/src/sage/interfaces/maxima_abstract.py b/src/sage/interfaces/maxima_abstract.py
        EXAMPLES::

            # The output is kind of random
            sage: sorted(maxima._commands(verbose=False))
            [...
             'display',
             ...
diff --git a/src/sage/manifolds/differentiable/affine_connection.py b/src/sage/manifolds/differentiable/affine_connection.py
            sage: ef = e.coframe()
            sage: ef[1].display(), ef[2].display(), ef[3].display()
            (e^1 = 1/y dx, e^2 = 1/z dy, e^3 = 1/x dz)
            sage: nab.curvature_form(1,1,e)  # long time
            2-form curvature (1,1) of connection nabla w.r.t. Vector frame
             (M, (e_1,e_2,e_3)) on the 3-dimensional differentiable manifold M
            sage: nab.curvature_form(1,1,e).display(e)  # long time (if above is skipped)
diff --git a/src/sage/plot/plot3d/transform.pyx b/src/sage/plot/plot3d/transform.pyx
        And simplify every single entry (which is more effective that simplify
        the whole matrix like above)::

            sage: m.parent()([x.simplify_full() for x in m._list()])  # random  # long time, needs sage.symbolic
            [                                       -(cos(theta) - 1)*x^2 + cos(theta)              -(cos(theta) - 1)*sqrt(-x^2 - z^2 + 1)*x + sin(theta)*abs(z)      -((cos(theta) - 1)*x*z^2 + sqrt(-x^2 - z^2 + 1)*sin(theta)*abs(z))/z]
            [             -(cos(theta) - 1)*sqrt(-x^2 - z^2 + 1)*x - sin(theta)*abs(z)                           (cos(theta) - 1)*x^2 + (cos(theta) - 1)*z^2 + 1 -((cos(theta) - 1)*sqrt(-x^2 - z^2 + 1)*z*abs(z) - x*z*sin(theta))/abs(z)]
            [     -((cos(theta) - 1)*x*z^2 - sqrt(-x^2 - z^2 + 1)*sin(theta)*abs(z))/z -((cos(theta) - 1)*sqrt(-x^2 - z^2 + 1)*z*abs(z) + x*z*sin(theta))/abs(z)                                        -(cos(theta) - 1)*z^2 + cos(theta)]
diff --git a/src/sage/rings/asymptotic/asymptotics_multivariate_generating_functions.py b/src/sage/rings/asymptotic/asymptotics_multivariate_generating_functions.py
            sage: asy # long time
            (4/3*sqrt(3)*sqrt(r)/sqrt(pi) + 47/216*sqrt(3)/(sqrt(pi)*sqrt(r)),
             1, 4/3*sqrt(3)*sqrt(r)/sqrt(pi) + 47/216*sqrt(3)/(sqrt(pi)*sqrt(r)))
            sage: F.relative_error(asy[0], alpha, [1, 2, 4, 8], asy[1]) # long time
            [((3, 3, 2), 0.9812164307, [1.515572606], [-0.54458543...]),
             ((6, 6, 4), 1.576181132, [1.992989399], [-0.26444185...]),
             ((12, 12, 8), 2.485286378, [2.712196351], [-0.091301338...]),
        sage: B = function('B')(*tuple(T))
        sage: AB_derivs = {}
        sage: M = matrix([[1, 2],[2, 1]])
        sage: DD = diff_op(A, B, AB_derivs, T, M, 1, 2)  # long time (see :issue:`35207`)
        sage: sorted(DD)                                 # long time
        [(0, 0, 0), (0, 1, 0), (0, 1, 1), (0, 1, 2)]
        sage: DD[(0, 1, 2)].number_of_operands()         # long time
diff --git a/src/sage/rings/asymptotic/asymptotic_expansion_generators.py b/src/sage/rings/asymptotic/asymptotic_expansion_generators.py

            sage: tau = var('tau')
            sage: phi = function('phi')
            sage: asymptotic_expansions.ImplicitExpansion('Z', phi=phi, tau=tau, precision=3)  # long time
            tau + (-sqrt(2)*sqrt(-tau*phi(tau)^2/(2*tau*diff(phi(tau), tau)^2
            - tau*phi(tau)*diff(phi(tau), tau, tau)
            - 2*phi(tau)*diff(phi(tau), tau))))*Z^(-1/2) + O(Z^(-1))
diff --git a/src/sage/rings/function_field/function_field.py b/src/sage/rings/function_field/function_field.py

    sage: TestSuite(J).run()
    sage: TestSuite(K).run(max_runs=256)        # long time (10s)                       # needs sage.rings.function_field sage.rings.number_field
    sage: TestSuite(L).run(max_runs=8)          # long time (25s)                       # needs sage.rings.function_field sage.rings.number_field

    sage: # needs sage.rings.finite_rings sage.rings.function_field
    sage: TestSuite(M).run(max_runs=8)                                  # long time (35s)
    sage: TestSuite(N).run(max_runs=8, skip='_test_derivation')         # long time (15s)
    sage: TestSuite(O).run()
    sage: TestSuite(R).run()
diff --git a/src/sage/rings/function_field/jacobian_khuri_makdisi.py b/src/sage/rings/function_field/jacobian_khuri_makdisi.py
        sage: J = C.jacobian(model='km_large', base_div=h)
        sage: G1 = J.group()
        sage: K = k.extension(2)
        sage: G2 = J.group(K)
        sage: G2.coerce_map_from(G1)
        Jacobian group embedding map:
          From: Group of rational points of Jacobian
        sage: J = C.jacobian(model='km_large', base_div=h)
        sage: G1 = J.group()
        sage: K = k.extension(2)
        sage: G2 = J.group(K)
        sage: G2.coerce_map_from(G1)
        Jacobian group embedding map:
          From: Group of rational points of Jacobian
            sage: b = C([0,0,1]).place()
            sage: J = C.jacobian(model='km_large', base_div=3*b)
            sage: G = J.group()
            sage: len([pt for pt in G])  # long time
            11
        """
        d0 = self._base_div.degree()
diff --git a/src/sage/rings/function_field/khuri_makdisi.pyx b/src/sage/rings/function_field/khuri_makdisi.pyx
            sage: D1 = 3*infty2 + infty1 - 4*P
            sage: D2 = F.divisor_group().zero()
            sage: J = F.jacobian(model='km-small', base_div=4*P)
            sage: J(D1) + J(D2) == J(D1)
            True
        """
        cdef int d0 = self.d0
diff --git a/src/sage/algebras/lie_algebras/classical_lie_algebra.py b/src/sage/algebras/lie_algebras/classical_lie_algebra.py
            sage: g
            Simple matrix Lie algebra of type ['E', 7] over Rational Field

            sage: len(g.basis())  # long time
            133
            sage: TestSuite(g).run()  # long time
        """
diff --git a/src/sage/algebras/quatalg/quaternion_algebra.py b/src/sage/algebras/quatalg/quaternion_algebra.py

        Check that randomly generated ideals decompose as expected::

            sage: for d in ( m for m in range(400, 750) if is_squarefree(m) ):        # long time (7s)
            ....:     A = QuaternionAlgebra(d)
            ....:     O = A.maximal_order()
            ....:     for _ in range(10):
diff --git a/src/sage/categories/super_hopf_algebras_with_basis.py b/src/sage/categories/super_hopf_algebras_with_basis.py
            TESTS::

                sage: A = SteenrodAlgebra(7)                                            # needs sage.combinat sage.modules
                sage: A._test_antipode()        # long time                             # needs sage.combinat sage.modules
            """
            tester = self._tester(**options)

diff --git a/src/sage/combinat/bijectionist.py b/src/sage/combinat/bijectionist.py
    sage: bij = Bijectionist(sum(As, []), sum(Bs, []))
    sage: bij.set_statistics((lambda x: x[0], lambda x: x[0]))
    sage: bij.set_intertwining_relations((2, c1, c1), (1, c2, c2))
    sage: l = list(bij.solutions_iterator()); len(l)                            # long time -- (2.7 seconds with SCIP on AMD Ryzen 5 PRO 3500U w/ Radeon Vega Mobile Gfx)
    64

 A brute force check would be difficult::
    sage: A = sum(As, [])
    sage: respects_c1 = lambda s: all(c1(a1, a2) not in A or s[c1(a1, a2)] == c1(s[a1], s[a2]) for a1 in A for a2 in A)
    sage: respects_c2 = lambda s: all(c2(a1) not in A or s[c2(a1)] == c2(s[a1]) for a1 in A)
    sage: l2 = [s for s in it if respects_c1(s) and respects_c2(s)]             # long time -- (17 seconds on AMD Ryzen 5 PRO 3500U w/ Radeon Vega Mobile Gfx)
    sage: sorted(l1, key=lambda s: tuple(s.items())) == l2                      # long time
    True

diff --git a/src/sage/combinat/backtrack.py b/src/sage/combinat/backtrack.py
    We factor out the long test from the ``TestSuite``::

        sage: TestSuite(PP).run(skip='_test_enumerated_set_contains')
        sage: PP._test_enumerated_set_contains()  # long time
    """

    def __init__(self):
diff --git a/src/sage/combinat/crystals/alcove_path.py b/src/sage/combinat/crystals/alcove_path.py
            sage: TestSuite(C).run()

            sage: C = crystals.AlcovePaths(['A',2,1],[1,0,0])
            sage: TestSuite(C).run() #long time

            sage: C = crystals.AlcovePaths(['A',2,1],[1,0],False)
            sage: TestSuite(C).run(skip='_test_stembridge_local_axioms') #long time
diff --git a/src/sage/combinat/cluster_algebra_quiver/quiver.py b/src/sage/combinat/cluster_algebra_quiver/quiver.py
            Quiver on 10 vertices of type ['E', 8, [1, 1]]
            sage: Q._mutation_type = None; Q
            Quiver on 10 vertices
            sage: Q.mutation_type() # long time
            ['E', 8, [1, 1]]

        the not yet working affine type D (unless user has saved small classical quiver data)::
            ['E', 8, [1, 1]]

            sage: dg = DiGraph( { 0:[3], 1:[0,4], 2:[0,6], 3:[1,2,7], 4:[3,8], 5:[2], 6:[3,5], 7:[4,6], 8:[7] } )
            sage: ClusterQuiver( dg ).mutation_type() # long time
            ['E', 8, 1]

            sage: dg = DiGraph( { 0:[3,9], 1:[0,4], 2:[0,6], 3:[1,2,7], 4:[3,8], 5:[2], 6:[3,5], 7:[4,6], 8:[7], 9:[1] } )
diff --git a/src/sage/combinat/cluster_algebra_quiver/cluster_seed.py b/src/sage/combinat/cluster_algebra_quiver/cluster_seed.py
            A seed for a cluster algebra of rank 10 of type ['E', 8, [1, 1]]
            sage: S._mutation_type = S._quiver._mutation_type = None; S
            A seed for a cluster algebra of rank 10
            sage: S.mutation_type() # long time
            ['E', 8, [1, 1]]

        - the not yet working affine type D::
diff --git a/src/sage/coding/linear_code.py b/src/sage/coding/linear_code.py

            sage: C = LinearCode(random_matrix(GF(47), 25, 35))
            sage: from sage.doctest.util import ensure_interruptible_after
            sage: with ensure_interruptible_after(0.5): C.canonical_representative()    # needs sage.libs.gap
        """
        aut_group_can_label = self._canonize(equivalence)
        return aut_group_can_label.get_canonical_form(), \
	These formulas are automatically used if we ask to plot the grid of spherical
	coordinates in terms of Cartesian coordinates::

	sage: spherical.plot(cartesian, color={r:'red', th:'green', ph:'orange'})
Check warning on line 141 in src/doc/en/thematic_tutorials/vector_calculus/vector_calc_change.rst GitHub Actions / Conda (ubuntu, Python 3.12) Warning: slow doctest: `slow doctest:` Check warning on line 141 in src/doc/en/thematic_tutorials/vector_calculus/vector_calc_change.rst GitHub Actions / Conda (ubuntu, Python 3.11) Warning: slow doctest: `slow doctest:` Check warning on line 141 in src/doc/en/thematic_tutorials/vector_calculus/vector_calc_change.rst GitHub Actions / Conda (ubuntu, Python 3.12, editable) Warning: slow doctest: `slow doctest:`
	Graphics3d Object

	.. PLOT::

	The Laplacian of a vector field::

	sage: Du = laplacian(u)
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	sage: Du.display()
	Delta(u) = ((r^2d^2(u_r)/dr^2 + 2rd(u_r)/dr - 2u_r(r, th, ph)
	+ d^2(u_r)/dth^2 - 2d(u_theta)/dth)sin(th)^2 - ((2*u_theta(r, th, ph)
	and also projective modules)::

	sage: SGA = SymmetricGroupAlgebra(QQ, 5)
	sage: for la in Partitions(SGA.n):
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	....: idem = SGA.ladder_idemponent(la)
	....: assert idem^2 == idem
	....: print(la, SGA.principal_ideal(idem).dimension())

	::

	sage: solution = H.self_surrounding(8, remove_incomplete_copies=False)
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	sage: G = sum([p.show2d() for p in solution], Graphics()) # needs sage.plot
	"""
	# Define the box to tile
	sage: W.start()
	sage: DC = DocTestController(DD, filename)
	sage: reporter = DocTestReporter(DC)
	sage: W.join() # Wait for worker to finish
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	sage: result = W.result_queue.get()
	sage: reporter.report(FDS, False, W.exitcode, result, "")
	[... tests, ...s wall]
	....: except OSError:
	....: print("Write end of pipe successfully closed")
	Write end of pipe successfully closed
	sage: W.join() # Wait for worker to finish
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	"""
	super().start()

	sage: W.start()
	sage: while W.rmessages is not None:
	....: W.read_messages()
	sage: W.join()
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	sage: len(W.messages) > 0
	True
	"""
	sage: FDS = FileDocTestSource(filename, DD)
	sage: W = DocTestWorker(FDS, DD)
	sage: W.start()
	sage: W.join()
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	sage: W.save_result_output()
	sage: sorted(W.result[1].keys())
	['cputime', 'err', 'failures', 'optionals', 'tests', 'walltime', 'walltime_skips']
	sage: P.<x,y>=ProjectiveSpace(QQbar, 1)
	sage: E=EllipticCurve([1, 2])
	sage: f=P.Lattes_map(E, 2)
	sage: f.Lattes_to_curve(check_lattes=true)
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	Elliptic Curve defined by y^2 = x^3 + x + 2 over Rational Field

	"""
	Exact value of the integral \(\displaystyle\int_{0}^{2}x^{2} +
	1\,\mathrm{d}x=4.666666666666668\)

	sage: test(interacts.calculus.function_tool)
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	...Interactive function <function function_tool at ...> with 7 widgets
	f: EvalText(value='sin(x)', description='f')
	g: EvalText(value='cos(x)', description='g')
	sage: from sage.combinat.designs.orthogonal_arrays_find_recursive import find_recursive_construction
	sage: from sage.combinat.designs.orthogonal_arrays import is_orthogonal_array
	sage: count = 0
	sage: for n in range(10,150):
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	....: k = designs.orthogonal_arrays.largest_available_k(n)
	....: if find_recursive_construction(k,n):
	....: count = count + 1
	True
	sage: F41 = FusionRing("F4", 1)
	sage: fmats = F41.get_fmatrix()
	sage: fmats.find_orthogonal_solution(verbose=False)
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	sage: b = F41.basis()
	sage: all(F41.s_ijconj(x, y) == F41._basecoer(F41.s_ij(x, y, base_coercion=False).conjugate()) for x in b for y in b)
	True
	sage: x = SR.var('x')
	sage: n = ZZ.random_element(5, 5001)
	sage: a = QQ.random_element().abs() + 5
	sage: s = ( (n + 1)*ultraspherical(n + 1, a, x)
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	....: - 2x(n + a)*ultraspherical(n, a, x)
	....: + (n + 2a - 1)ultraspherical(n - 1, a, x) )
	sage: s.expand().is_zero()