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Matthias Koeppe
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Update # optional / # needs
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‎src/sage/rings/polynomial/infinite_polynomial_element.py

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@@ -48,8 +48,8 @@
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There is a permutation action on Infinite Polynomial Rings by
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permuting the indices of the variables::
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sage: P = Permutation(((4,5),(2,3))) # optional - sage.combinat
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sage: c^P # optional - sage.combinat
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sage: P = Permutation(((4,5),(2,3)))
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sage: c^P
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x_2^3 + x_2*y_5 - 2*y_5^4
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Note that ``P(0)==0``, and thus variables of index zero are invariant
@@ -440,13 +440,13 @@ def subs(self, fixed=None, **kwargs):
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The substitution can also handle matrices::
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sage: M = matrix([[1,0], [0,2]]) # optional - sage.modules
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sage: N = matrix([[0,3], [4,0]]) # optional - sage.modules
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sage: g = x[0]^2 + 3*x[1] # optional - sage.modules
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sage: g.subs({'x_0': M}) # optional - sage.modules
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sage: M = matrix([[1,0], [0,2]]) # needs sage.modules
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sage: N = matrix([[0,3], [4,0]]) # needs sage.modules
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sage: g = x[0]^2 + 3*x[1] # needs sage.modules
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sage: g.subs({'x_0': M}) # needs sage.modules
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[3*x_1 + 1 0]
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[ 0 3*x_1 + 4]
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sage: g.subs({x[0]: M, x[1]: N}) # optional - sage.modules
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sage: g.subs({x[0]: M, x[1]: N}) # needs sage.modules
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[ 1 9]
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[12 4]
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@@ -464,7 +464,7 @@ def subs(self, fixed=None, **kwargs):
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TESTS::
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sage: g.subs(fixed=x[0], x_1=N)
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sage: g.subs(fixed=x[0], x_1=N) # needs sage.modules
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Traceback (most recent call last):
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...
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ValueError: fixed must be a dict
@@ -540,10 +540,10 @@ def is_nilpotent(self):
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EXAMPLES::
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sage: R.<x> = InfinitePolynomialRing(QQbar) # optional - sage.rings.number_field
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sage: (x[0] + x[1]).is_nilpotent() # optional - sage.rings.number_field
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sage: R.<x> = InfinitePolynomialRing(QQbar) # needs sage.rings.number_field
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sage: (x[0] + x[1]).is_nilpotent() # needs sage.rings.number_field
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False
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sage: R(0).is_nilpotent() # optional - sage.rings.number_field
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sage: R(0).is_nilpotent() # needs sage.rings.number_field
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True
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sage: _.<x> = InfinitePolynomialRing(Zmod(4))
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sage: (2*x[0]).is_nilpotent()
@@ -656,7 +656,7 @@ def _div_(self, x):
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sage: z = 1/(x[2]*(x[1]+x[2]))+1/(x[1]*(x[1]+x[2]))
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sage: z.parent()
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Fraction Field of Infinite polynomial ring in x over Rational Field
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sage: factor(z) # optional - sage.libs.singular
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sage: factor(z) # needs sage.libs.singular
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x_1^-1 * x_2^-1
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"""
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if not x.variables():
@@ -896,11 +896,11 @@ def symmetric_cancellation_order(self, other):
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sage: X.<x,y> = InfinitePolynomialRing(QQ)
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sage: (x[2]*x[1]).symmetric_cancellation_order(x[2]^2)
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(None, 1, 1)
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sage: (x[2]*x[1]).symmetric_cancellation_order(x[2]*x[3]*y[1]) # optional - sage.combinat
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sage: (x[2]*x[1]).symmetric_cancellation_order(x[2]*x[3]*y[1])
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(-1, [2, 3, 1], y_1)
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sage: (x[2]*x[1]*y[1]).symmetric_cancellation_order(x[2]*x[3]*y[1]) # optional - sage.combinat
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sage: (x[2]*x[1]*y[1]).symmetric_cancellation_order(x[2]*x[3]*y[1])
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(None, 1, 1)
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sage: (x[2]*x[1]*y[1]).symmetric_cancellation_order(x[2]*x[3]*y[2]) # optional - sage.combinat
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sage: (x[2]*x[1]*y[1]).symmetric_cancellation_order(x[2]*x[3]*y[2])
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(-1, [2, 3, 1], 1)
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"""
@@ -1097,23 +1097,23 @@ def reduce(self, I, tailreduce=False, report=None):
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reduction. However, reduction by ``y[1]*x[2]^2`` works, since
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one can change variable index 1 into 2 and 2 into 3::
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sage: p.reduce([y[1]*x[2]^2]) # optional - sage.libs.singular
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sage: p.reduce([y[1]*x[2]^2]) # needs sage.libs.singular
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y_3*y_1^2
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The next example shows that tail reduction is not done, unless
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it is explicitly advised. The input can also be a Symmetric
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Ideal::
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sage: I = (y[3])*X
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sage: p.reduce(I) # optional - sage.libs.singular
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sage: p.reduce(I)
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x_3^3*y_2 + y_3*y_1^2
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sage: p.reduce(I, tailreduce=True) # optional - sage.libs.singular
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sage: p.reduce(I, tailreduce=True) # needs sage.libs.singular
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x_3^3*y_2
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Last, we demonstrate the ``report`` option::
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sage: p = x[1]^2 + y[2]^2 + x[1]*x[2]*y[3] + x[1]*y[4]
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sage: p.reduce(I, tailreduce=True, report=True) # optional - sage.libs.singular
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sage: p.reduce(I, tailreduce=True, report=True) # needs sage.libs.singular
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:T[2]:>
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>
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x_1^2 + y_2^2
@@ -1258,8 +1258,8 @@ def __call__(self, *args, **kwargs):
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sage: a(x_1=x[100])
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x_100 + x_0
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sage: M = matrix([[1,1], [2,0]]) # optional - sage.modules
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sage: a(x_1=M) # optional - sage.modules
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sage: M = matrix([[1,1], [2,0]]) # needs sage.modules
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sage: a(x_1=M) # needs sage.modules
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[x_0 + 1 1]
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[ 2 x_0]
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"""
@@ -1389,8 +1389,8 @@ def __pow__(self, n):
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sage: X.<x,y> = InfinitePolynomialRing(QQ, implementation='sparse')
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sage: p = x[10]*y[2] + 2*x[1]*y[3]
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sage: P = Permutation(((1,2),(3,4,5))) # optional - sage.combinat
1393-
sage: p^P # indirect doctest # optional - sage.combinat
1392+
sage: P = Permutation(((1,2),(3,4,5)))
1393+
sage: p^P # indirect doctest
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x_10*y_1 + 2*x_2*y_4
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"""
@@ -1665,8 +1665,8 @@ def __pow__(self, n):
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sage: x[10]^3
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x_10^3
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sage: p = x[10]*y[2] + 2*x[1]*y[3]
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sage: P = Permutation(((1,2),(3,4,5))) # optional - sage.combinat
1669-
sage: p^P # optional - sage.combinat
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sage: P = Permutation(((1,2),(3,4,5)))
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sage: p^P
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x_10*y_1 + 2*x_2*y_4
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"""

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