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sage.rings.polynomial: More block tags, doctest cosmetics
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Matthias Koeppe committed Aug 8, 2023
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4 changes: 2 additions & 2 deletions src/sage/rings/polynomial/multi_polynomial_element.py
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Original file line number Diff line number Diff line change
Expand Up @@ -550,7 +550,7 @@ def _macaulay2_(self, macaulay2=None):
EXAMPLES::
sage: R = GF(13)['a,b']['c,d']
sage: macaulay2(R('a^2 + c')) # optional - macaulay2, needs sage.rings.finite_rings
sage: macaulay2(R('a^2 + c')) # optional - macaulay2
2
c + a
Expand All @@ -559,7 +559,7 @@ def _macaulay2_(self, macaulay2=None):
Elements of the base ring are coerced to the polynomial ring
correctly::
sage: macaulay2(R('a^2')).ring()._operator('===', R) # optional - macaulay2, needs sage.rings.finite_rings
sage: macaulay2(R('a^2')).ring()._operator('===', R) # optional - macaulay2
true
"""
if macaulay2 is None:
Expand Down
83 changes: 41 additions & 42 deletions src/sage/rings/polynomial/multi_polynomial_ideal.py
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Original file line number Diff line number Diff line change
Expand Up @@ -347,9 +347,10 @@ def _magma_init_(self, magma):
EXAMPLES::
sage: R.<a,b,c,d,e,f,g,h,i,j> = PolynomialRing(GF(127),10)
sage: I = sage.rings.ideal.Cyclic(R,4) # indirect doctest # needs sage.rings.finite_rings
sage: magma(I) # optional - magma # needs sage.rings.finite_rings
sage: # optional - magma
sage: R.<a,b,c,d,e,f,g,h,i,j> = PolynomialRing(GF(127), 10)
sage: I = sage.rings.ideal.Cyclic(R,4) # indirect doctest
sage: magma(I)
Ideal of Polynomial ring of rank 10 over GF(127)
Order: Graded Reverse Lexicographical
Variables: a, b, c, d, e, f, g, h, i, j
Expand Down Expand Up @@ -384,20 +385,20 @@ def _groebner_basis_magma(self, deg_bound=None, prot=False, magma=magma_default)
EXAMPLES::
sage: # needs sage.rings.finite_rings
sage: # optional - magma
sage: R.<a,b,c,d,e,f,g,h,i,j> = PolynomialRing(GF(127), 10)
sage: I = sage.rings.ideal.Cyclic(R, 6)
sage: gb = I.groebner_basis('magma:GroebnerBasis') # optional - magma
sage: len(gb) # optional - magma
sage: gb = I.groebner_basis('magma:GroebnerBasis')
sage: len(gb)
45
We may also pass a degree bound to Magma::
sage: # needs sage.rings.finite_rings
sage: # optional - magma
sage: R.<a,b,c,d,e,f,g,h,i,j> = PolynomialRing(GF(127), 10)
sage: I = sage.rings.ideal.Cyclic(R, 6)
sage: gb = I.groebner_basis('magma:GroebnerBasis', deg_bound=4) # optional - magma
sage: len(gb) # optional - magma
sage: gb = I.groebner_basis('magma:GroebnerBasis', deg_bound=4)
sage: len(gb)
5
"""
R = self.ring()
Expand Down Expand Up @@ -1354,14 +1355,14 @@ def _groebner_basis_ginv(self, algorithm="TQ", criteria='CritPartially', divisio
Currently, only `\GF{p}` and `\QQ` are supported as base fields::
sage: P.<x,y,z> = PolynomialRing(QQ,order='degrevlex')
sage: # optional - ginv
sage: P.<x,y,z> = PolynomialRing(QQ, order='degrevlex')
sage: I = sage.rings.ideal.Katsura(P)
sage: I.groebner_basis(algorithm='ginv') # optional - ginv
sage: I.groebner_basis(algorithm='ginv')
[z^3 - 79/210*z^2 + 1/30*y + 1/70*z, y^2 - 3/5*z^2 - 1/5*y + 1/5*z, y*z + 6/5*z^2 - 1/10*y - 2/5*z, x + 2*y + 2*z - 1]
sage: P.<x,y,z> = PolynomialRing(GF(127), order='degrevlex')
sage: I = sage.rings.ideal.Katsura(P) # needs sage.rings.finite_rings
sage: I.groebner_basis(algorithm='ginv') # optional - ginv # needs sage.rings.finite_rings
sage: I = sage.rings.ideal.Katsura(P)
sage: I.groebner_basis(algorithm='ginv')
...
[z^3 + 22*z^2 - 55*y + 49*z, y^2 - 26*z^2 - 51*y + 51*z, y*z + 52*z^2 + 38*y + 25*z, x + 2*y + 2*z - 1]
Expand Down Expand Up @@ -1979,7 +1980,7 @@ def interreduced_basis(self):
The interreduced basis of 0 is 0::
sage: P.<x,y,z> = GF(2)[]
sage: Ideal(P(0)).interreduced_basis() # needs sage.rings.finite_rings
sage: Ideal(P(0)).interreduced_basis()
[0]
ALGORITHM:
Expand Down Expand Up @@ -2526,8 +2527,6 @@ def variety(self, ring=None, *, algorithm="triangular_decomposition", proof=True
y^48 + y^41 - y^40 + y^37 - y^36 - y^33 + y^32 - y^29 + y^28
- y^25 + y^24 + y^2 + y + 1)
of Multivariate Polynomial Ring in x, y over Finite Field in w of size 3^3
sage: # needs sage.rings.finite_rings
sage: V = I.variety();
sage: sorted(V, key=str)
[{y: w^2 + 2*w, x: 2*w + 2}, {y: w^2 + 2, x: 2*w}, {y: w^2 + w, x: 2*w + 1}]
Expand Down Expand Up @@ -2610,9 +2609,10 @@ def variety(self, ring=None, *, algorithm="triangular_decomposition", proof=True
If the ground field's characteristic is too large for
Singular, we resort to a toy implementation::
sage: R.<x,y> = PolynomialRing(GF(2147483659^3), order='lex') # needs sage.rings.finite_rings
sage: I = ideal([x^3 - 2*y^2, 3*x + y^4]) # needs sage.rings.finite_rings
sage: I.variety() # needs sage.rings.finite_rings
sage: # needs sage.rings.finite_rings
sage: R.<x,y> = PolynomialRing(GF(2147483659^3), order='lex')
sage: I = ideal([x^3 - 2*y^2, 3*x + y^4])
sage: I.variety()
verbose 0 (...: multi_polynomial_ideal.py, groebner_basis) Warning: falling back to very slow toy implementation.
verbose 0 (...: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.
verbose 0 (...: multi_polynomial_ideal.py, variety) Warning: falling back to very slow toy implementation.
Expand All @@ -2623,22 +2623,23 @@ def variety(self, ring=None, *, algorithm="triangular_decomposition", proof=True
But the mapping will also accept generators of the original ring,
or even generator names as strings, when provided as keys::
sage: # needs sage.rings.number_field
sage: K.<x,y> = QQ[]
sage: I = ideal([x^2 + 2*y - 5, x + y + 3])
sage: v = I.variety(AA)[0]; v[x], v[y] # needs sage.rings.number_field
sage: v = I.variety(AA)[0]; v[x], v[y]
(4.464101615137755?, -7.464101615137755?)
sage: list(v)[0].parent() # needs sage.rings.number_field
sage: list(v)[0].parent()
Multivariate Polynomial Ring in x, y over Algebraic Real Field
sage: v[x] # needs sage.rings.number_field
sage: v[x]
4.464101615137755?
sage: v["y"] # needs sage.rings.number_field
sage: v["y"]
-7.464101615137755?
``msolve`` also works over finite fields::
sage: R.<x, y> = PolynomialRing(GF(536870909), 2, order='lex') # needs sage.rings.finite_rings
sage: I = Ideal([x^2 - 1, y^2 - 1]) # needs sage.rings.finite_rings
sage: sorted(I.variety(algorithm='msolve', # optional - msolve # needs sage.rings.finite_rings
sage: sorted(I.variety(algorithm='msolve', # optional - msolve, needs sage.rings.finite_rings
....: proof=False),
....: key=str)
[{x: 1, y: 1},
Expand All @@ -2651,7 +2652,7 @@ def variety(self, ring=None, *, algorithm="triangular_decomposition", proof=True
sage: R.<x, y> = PolynomialRing(GF(3), 2, order='lex')
sage: I = Ideal([x^2 - 1, y^2 - 1])
sage: I.variety(algorithm='msolve', proof=False) # optional - msolve, needs sage.rings.finite_rings
sage: I.variety(algorithm='msolve', proof=False) # optional - msolve
Traceback (most recent call last):
...
NotImplementedError: characteristic 3 too small
Expand Down Expand Up @@ -2710,10 +2711,10 @@ def _variety_triangular_decomposition(self, ring):
x11^2 + x11, x12^2 + x12, x13^2 + x13, x14^2 + x14, x15^2 + x15, \
x16^2 + x16, x17^2 + x17, x18^2 + x18, x19^2 + x19, x20^2 + x20, \
x21^2 + x21, x22^2 + x22, x23^2 + x23, x24^2 + x24, x25^2 + x25, \
x26^2 + x26, x27^2 + x27, x28^2 + x28, x29^2 + x29, x30^2 + x30]) # optional - sage.rings.finite_rings
sage: I.basis_is_groebner() # needs sage.rings.finite_rings
x26^2 + x26, x27^2 + x27, x28^2 + x28, x29^2 + x29, x30^2 + x30])
sage: I.basis_is_groebner()
True
sage: sorted("".join(str(V[g]) for g in R.gens()) for V in I.variety()) # long time (6s on sage.math, 2011), needs sage.rings.finite_rings
sage: sorted("".join(str(V[g]) for g in R.gens()) for V in I.variety()) # long time (6s on sage.math, 2011)
['101000100000000110001000100110',
'101000100000000110001000101110',
'101000100100000101001000100110',
Expand Down Expand Up @@ -2748,10 +2749,10 @@ def _variety_triangular_decomposition(self, ring):
Check that the issue at :trac:`7425` is fixed::
sage: S.<t>=PolynomialRing(QQ)
sage: F.<q>=QQ.extension(t^4+1)
sage: R.<x,y>=PolynomialRing(F)
sage: I=R.ideal(x,y^4+1)
sage: S.<t> = PolynomialRing(QQ)
sage: F.<q> = QQ.extension(t^4 + 1)
sage: R.<x,y> = PolynomialRing(F)
sage: I = R.ideal(x, y^4 + 1)
sage: I.variety()
[...{y: -q^3, x: 0}...]
Expand All @@ -2765,7 +2766,7 @@ def _variety_triangular_decomposition(self, ring):
Check that the issue at :trac:`16485` is fixed::
sage: R.<a,b,c> = PolynomialRing(QQ, order='lex')
sage: I = R.ideal(c^2-2, b-c, a)
sage: I = R.ideal(c^2 - 2, b - c, a)
sage: I.variety(QQbar) # needs sage.rings.number_field
[...a: 0...]
Expand Down Expand Up @@ -4649,10 +4650,10 @@ def groebner_basis(self, algorithm='', deg_bound=None, mult_bound=None, prot=Fal
if not algorithm:
try:
gb = self._groebner_basis_libsingular("groebner", deg_bound=deg_bound, mult_bound=mult_bound, *args, **kwds)
except (TypeError, NameError, ImportError): # conversion to Singular not supported
except (TypeError, NameError, ImportError): # conversion to Singular not supported
try:
gb = self._groebner_basis_singular("groebner", deg_bound=deg_bound, mult_bound=mult_bound, *args, **kwds)
except (TypeError, NameError, NotImplementedError, ImportError): # conversion to Singular not supported
except (TypeError, NameError, NotImplementedError, ImportError): # conversion to Singular not supported
R = self.ring()
B = R.base_ring()
if R.ngens() == 0:
Expand Down Expand Up @@ -5420,8 +5421,7 @@ def weil_restriction(self):
sage: J += sage.rings.ideal.FieldIdeal(J.ring()) # ensure radical ideal
sage: J.variety()
[{y1: 1, y0: 0, x1: 1, x0: 1}]
sage: J.weil_restriction() # returns J # needs sage.rings.finite_rings
sage: J.weil_restriction() # returns J
Ideal (x0*y0 + x1*y1 + 1, x1*y0 + x0*y1 + x1*y1, x1 + 1, x0 + x1,
x0^2 + x0, x1^2 + x1, y0^2 + y0, y1^2 + y1) of Multivariate
Polynomial Ring in x0, x1, y0, y1 over Finite Field of size 2
Expand All @@ -5434,8 +5434,7 @@ def weil_restriction(self):
0
sage: I.variety()
[{z: 0, y: 0, x: 1}]
sage: J = I.weil_restriction(); J # needs sage.rings.finite_rings
sage: J = I.weil_restriction(); J
Ideal (x0 - y0 - z0 - 1,
x1 - y1 - z1, x2 - y2 - z2, x3 - y3 - z3, x4 - y4 - z4,
x0^2 + x2*x3 + x1*x4 - y0^2 - y2*y3 - y1*y4 - z0^2 - z2*z3 - z1*z4 - x0,
Expand All @@ -5460,9 +5459,9 @@ def weil_restriction(self):
- y4*z0 - y3*z1 - y2*z2 - y1*z3 - y0*z4 - y4*z4 - y4)
of Multivariate Polynomial Ring in x0, x1, x2, x3, x4, y0, y1, y2, y3, y4,
z0, z1, z2, z3, z4 over Finite Field of size 3
sage: J += sage.rings.ideal.FieldIdeal(J.ring()) # ensure radical ideal # needs sage.rings.finite_rings
sage: J += sage.rings.ideal.FieldIdeal(J.ring()) # ensure radical ideal
sage: from sage.doctest.fixtures import reproducible_repr
sage: print(reproducible_repr(J.variety())) # needs sage.rings.finite_rings
sage: print(reproducible_repr(J.variety()))
[{x0: 1, x1: 0, x2: 0, x3: 0, x4: 0,
y0: 0, y1: 0, y2: 0, y3: 0, y4: 0,
z0: 0, z1: 0, z2: 0, z3: 0, z4: 0}]
Expand Down
2 changes: 1 addition & 1 deletion src/sage/rings/polynomial/multi_polynomial_libsingular.pyx
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Original file line number Diff line number Diff line change
Expand Up @@ -5080,7 +5080,7 @@ cdef class MPolynomial_libsingular(MPolynomial_libsingular_base):
sage: # optional - macaulay2
sage: R.<x,y> = PolynomialRing(GF(7), 2)
sage: f = (x^3 + 2*y^2*x)^7; f
sage: f = (x^3 + 2*y^2*x)^7; f # indirect doctest
x^21 + 2*x^7*y^14
sage: h = macaulay2(f); h
21 7 14
Expand Down
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