Term orders¶
Sage supports the following term orders:
- Lexicographic (lex)
\(x^a < x^b\) if and only if there exists \(1 \le i \le n\) such that \(a_1 = b_1, \dots, a_{i-1} = b_{i-1}, a_i < b_i\). This term order is called ‘lp’ in Singular.
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQ, 3, order='lex') sage: x > y True sage: x > y^2 True sage: x > 1 True sage: x^1*y^2 > y^3*z^4 True sage: x^3*y^2*z^4 < x^3*y^2*z^1 False
>>> from sage.all import * >>> P = PolynomialRing(QQ, Integer(3), order='lex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> x > y True >>> x > y**Integer(2) True >>> x > Integer(1) True >>> x**Integer(1)*y**Integer(2) > y**Integer(3)*z**Integer(4) True >>> x**Integer(3)*y**Integer(2)*z**Integer(4) < x**Integer(3)*y**Integer(2)*z**Integer(1) False
- Degree reverse lexicographic (degrevlex)
Let \(\deg(x^a) = a_1 + a_2 + \dots + a_n\), then \(x^a < x^b\) if and only if \(\deg(x^a) < \deg(x^b)\) or \(\deg(x^a) = \deg(x^b)\) and there exists \(1 \le i \le n\) such that \(a_n = b_n, \dots, a_{i+1} = b_{i+1}, a_i > b_i\). This term order is called ‘dp’ in Singular.
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQ, 3, order='degrevlex') sage: x > y True sage: x > y^2*z False sage: x > 1 True sage: x^1*y^5*z^2 > x^4*y^1*z^3 True sage: x^2*y*z^2 > x*y^3*z False
>>> from sage.all import * >>> P = PolynomialRing(QQ, Integer(3), order='degrevlex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> x > y True >>> x > y**Integer(2)*z False >>> x > Integer(1) True >>> x**Integer(1)*y**Integer(5)*z**Integer(2) > x**Integer(4)*y**Integer(1)*z**Integer(3) True >>> x**Integer(2)*y*z**Integer(2) > x*y**Integer(3)*z False
- Degree lexicographic (deglex)
Let \(\deg(x^a) = a_1 + a_2 + \dots + a_n\), then \(x^a < x^b\) if and only if \(\deg(x^a) < \deg(x^b)\) or \(\deg(x^a) = \deg(x^b)\) and there exists \(1 \le i \le n\) such that \(a_1 = b_1, \dots, a_{i-1} = b_{i-1}, a_i < b_i\). This term order is called ‘Dp’ in Singular.
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQ, 3, order='deglex') sage: x > y True sage: x > y^2*z False sage: x > 1 True sage: x^1*y^2*z^3 > x^3*y^2*z^0 True sage: x^2*y*z^2 > x*y^3*z True
>>> from sage.all import * >>> P = PolynomialRing(QQ, Integer(3), order='deglex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> x > y True >>> x > y**Integer(2)*z False >>> x > Integer(1) True >>> x**Integer(1)*y**Integer(2)*z**Integer(3) > x**Integer(3)*y**Integer(2)*z**Integer(0) True >>> x**Integer(2)*y*z**Integer(2) > x*y**Integer(3)*z True
- Inverse lexicographic (invlex)
\(x^a < x^b\) if and only if there exists \(1 \le i \le n\) such that \(a_n = b_n, \dots, a_{i+1} = b_{i+1}, a_i < b_i\). This order is called ‘rp’ in Singular.
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQ, 3, order='invlex') sage: x > y False sage: y > x^2 True sage: x > 1 True sage: x*y > z False
>>> from sage.all import * >>> P = PolynomialRing(QQ, Integer(3), order='invlex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> x > y False >>> y > x**Integer(2) True >>> x > Integer(1) True >>> x*y > z False
This term order only makes sense in a non-commutative setting because if P is the ring \(k[x_1, \dots, x_n]\) and term order ‘invlex’ then it is equivalent to the ring \(k[x_n, \dots, x_1]\) with term order ‘lex’.
- Negative lexicographic (neglex)
\(x^a < x^b\) if and only if there exists \(1 \le i \le n\) such that \(a_1 = b_1, \dots, a_{i-1} = b_{i-1}, a_i > b_i\). This term order is called ‘ls’ in Singular.
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQ, 3, order='neglex') sage: x > y False sage: x > 1 False sage: x^1*y^2 > y^3*z^4 False sage: x^3*y^2*z^4 < x^3*y^2*z^1 True sage: x*y > z False
>>> from sage.all import * >>> P = PolynomialRing(QQ, Integer(3), order='neglex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> x > y False >>> x > Integer(1) False >>> x**Integer(1)*y**Integer(2) > y**Integer(3)*z**Integer(4) False >>> x**Integer(3)*y**Integer(2)*z**Integer(4) < x**Integer(3)*y**Integer(2)*z**Integer(1) True >>> x*y > z False
- Negative degree reverse lexicographic (negdegrevlex)
Let \(\deg(x^a) = a_1 + a_2 + \dots + a_n\), then \(x^a < x^b\) if and only if \(\deg(x^a) > \deg(x^b)\) or \(\deg(x^a) = \deg(x^b)\) and there exists \(1 \le i \le n\) such that \(a_n = b_n, \dots, a_{i+1} = b_{i+1}, a_i > b_i\). This term order is called ‘ds’ in Singular.
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQ, 3, order='negdegrevlex') sage: x > y True sage: x > x^2 True sage: x > 1 False sage: x^1*y^2 > y^3*z^4 True sage: x^2*y*z^2 > x*y^3*z False
>>> from sage.all import * >>> P = PolynomialRing(QQ, Integer(3), order='negdegrevlex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> x > y True >>> x > x**Integer(2) True >>> x > Integer(1) False >>> x**Integer(1)*y**Integer(2) > y**Integer(3)*z**Integer(4) True >>> x**Integer(2)*y*z**Integer(2) > x*y**Integer(3)*z False
- Negative degree lexicographic (negdeglex)
Let \(\deg(x^a) = a_1 + a_2 + \dots + a_n\), then \(x^a < x^b\) if and only if \(\deg(x^a) > \deg(x^b)\) or \(\deg(x^a) = \deg(x^b)\) and there exists \(1 \le i \le n\) such that \(a_1 = b_1, \dots, a_{i-1} = b_{i-1}, a_i < b_i\). This term order is called ‘Ds’ in Singular.
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQ, 3, order='negdeglex') sage: x > y True sage: x > x^2 True sage: x > 1 False sage: x^1*y^2 > y^3*z^4 True sage: x^2*y*z^2 > x*y^3*z True
>>> from sage.all import * >>> P = PolynomialRing(QQ, Integer(3), order='negdeglex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> x > y True >>> x > x**Integer(2) True >>> x > Integer(1) False >>> x**Integer(1)*y**Integer(2) > y**Integer(3)*z**Integer(4) True >>> x**Integer(2)*y*z**Integer(2) > x*y**Integer(3)*z True
- Weighted degree reverse lexicographic (wdegrevlex), positive integral weights
Let \(\deg_w(x^a) = a_1w_1 + a_2w_2 + \dots + a_nw_n\) with weights \(w\), then \(x^a < x^b\) if and only if \(\deg_w(x^a) < \deg_w(x^b)\) or \(\deg_w(x^a) = \deg_w(x^b)\) and there exists \(1 \le i \le n\) such that \(a_n = b_n, \dots, a_{i+1} = b_{i+1}, a_i > b_i\). This term order is called ‘wp’ in Singular.
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQ, 3, order=TermOrder('wdegrevlex',(1,2,3))) sage: x > y False sage: x > x^2 False sage: x > 1 True sage: x^1*y^2 > x^2*z True sage: y*z > x^3*y False
>>> from sage.all import * >>> P = PolynomialRing(QQ, Integer(3), order=TermOrder('wdegrevlex',(Integer(1),Integer(2),Integer(3))), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> x > y False >>> x > x**Integer(2) False >>> x > Integer(1) True >>> x**Integer(1)*y**Integer(2) > x**Integer(2)*z True >>> y*z > x**Integer(3)*y False
- Weighted degree lexicographic (wdeglex), positive integral weights
Let \(\deg_w(x^a) = a_1w_1 + a_2w_2 + \dots + a_nw_n\) with weights \(w\), then \(x^a < x^b\) if and only if \(\deg_w(x^a) < \deg_w(x^b)\) or \(\deg_w(x^a) = \deg_w(x^b)\) and there exists \(1 \le i \le n\) such that \(a_1 = b_1, \dots, a_{i-1} = b_{i-1}, a_i < b_i\). This term order is called ‘Wp’ in Singular.
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQ, 3, order=TermOrder('wdeglex',(1,2,3))) sage: x > y False sage: x > x^2 False sage: x > 1 True sage: x^1*y^2 > x^2*z False sage: y*z > x^3*y False
>>> from sage.all import * >>> P = PolynomialRing(QQ, Integer(3), order=TermOrder('wdeglex',(Integer(1),Integer(2),Integer(3))), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> x > y False >>> x > x**Integer(2) False >>> x > Integer(1) True >>> x**Integer(1)*y**Integer(2) > x**Integer(2)*z False >>> y*z > x**Integer(3)*y False
- Negative weighted degree reverse lexicographic (negwdegrevlex), positive integral weights
Let \(\deg_w(x^a) = a_1w_1 + a_2w_2 + \dots + a_nw_n\) with weights \(w\), then \(x^a < x^b\) if and only if \(\deg_w(x^a) > \deg_w(x^b)\) or \(\deg_w(x^a) = \deg_w(x^b)\) and there exists \(1 \le i \le n\) such that \(a_n = b_n, \dots, a_{i+1} = b_{i+1}, a_i > b_i\). This term order is called ‘ws’ in Singular.
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQ, 3, order=TermOrder('negwdegrevlex',(1,2,3))) sage: x > y True sage: x > x^2 True sage: x > 1 False sage: x^1*y^2 > x^2*z True sage: y*z > x^3*y False
>>> from sage.all import * >>> P = PolynomialRing(QQ, Integer(3), order=TermOrder('negwdegrevlex',(Integer(1),Integer(2),Integer(3))), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> x > y True >>> x > x**Integer(2) True >>> x > Integer(1) False >>> x**Integer(1)*y**Integer(2) > x**Integer(2)*z True >>> y*z > x**Integer(3)*y False
- Degree negative lexicographic (degneglex)
Let \(\deg(x^a) = a_1 + a_2 + \dots + a_n\), then \(x^a < x^b\) if and only if \(\deg(x^a) < \deg(x^b)\) or \(\deg(x^a) = \deg(x^b)\) and there exists \(1 \le i \le n\) such that \(a_1 = b_1, \dots, a_{i-1} = b_{i-1}, a_i > b_i\). This term order is called ‘dp_asc’ in PolyBoRi. Singular has the extra weight vector ordering
(a(1:n),ls)for this purpose.EXAMPLES:
sage: t = TermOrder('degneglex') sage: P.<x,y,z> = PolynomialRing(QQ, order=t) sage: x*y > y*z # indirect doctest False sage: x*y > x True
>>> from sage.all import * >>> t = TermOrder('degneglex') >>> P = PolynomialRing(QQ, order=t, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> x*y > y*z # indirect doctest False >>> x*y > x True
- Negative weighted degree lexicographic (negwdeglex), positive integral weights
Let \(\deg_w(x^a) = a_1w_1 + a_2w_2 + \dots + a_nw_n\) with weights \(w\), then \(x^a < x^b\) if and only if \(\deg_w(x^a) > \deg_w(x^b)\) or \(\deg_w(x^a) = \deg_w(x^b)\) and there exists \(1 \le i \le n\) such that \(a_1 = b_1, \dots, a_{i-1} = b_{i-1}, a_i < b_i\). This term order is called ‘Ws’ in Singular.
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQ, 3, order=TermOrder('negwdeglex',(1,2,3))) sage: x > y True sage: x > x^2 True sage: x > 1 False sage: x^1*y^2 > x^2*z False sage: y*z > x^3*y False
>>> from sage.all import * >>> P = PolynomialRing(QQ, Integer(3), order=TermOrder('negwdeglex',(Integer(1),Integer(2),Integer(3))), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> x > y True >>> x > x**Integer(2) True >>> x > Integer(1) False >>> x**Integer(1)*y**Integer(2) > x**Integer(2)*z False >>> y*z > x**Integer(3)*y False
Of these, only ‘degrevlex’, ‘deglex’, ‘degneglex’, ‘wdegrevlex’, ‘wdeglex’, ‘invlex’ and ‘lex’ are global orders.
Sage also supports matrix term order. Given a square matrix \(A\),
\(x^a <_A x^b\) if and only if \(Aa < Ab\)
where \(<\) is the lexicographic term order.
EXAMPLES:
sage: # needs sage.modules
sage: m = matrix(2, [2,3,0,1]); m
[2 3]
[0 1]
sage: T = TermOrder(m); T
Matrix term order with matrix
[2 3]
[0 1]
sage: P.<a,b> = PolynomialRing(QQ, 2, order=T)
sage: P
Multivariate Polynomial Ring in a, b over Rational Field
sage: a > b
False
sage: a^3 < b^2
True
sage: S = TermOrder('M(2,3,0,1)')
sage: T == S
True
>>> from sage.all import *
>>> # needs sage.modules
>>> m = matrix(Integer(2), [Integer(2),Integer(3),Integer(0),Integer(1)]); m
[2 3]
[0 1]
>>> T = TermOrder(m); T
Matrix term order with matrix
[2 3]
[0 1]
>>> P = PolynomialRing(QQ, Integer(2), order=T, names=('a', 'b',)); (a, b,) = P._first_ngens(2)
>>> P
Multivariate Polynomial Ring in a, b over Rational Field
>>> a > b
False
>>> a**Integer(3) < b**Integer(2)
True
>>> S = TermOrder('M(2,3,0,1)')
>>> T == S
True
Additionally all these monomial orders may be combined to product or block orders, defined as:
Let \(x = (x_1, x_2, \dots, x_n)\) and \(y = (y_1, y_2, \dots, y_m)\) be two ordered sets of variables, \(<_1\) a monomial order on \(k[x]\) and \(<_2\) a monomial order on \(k[y]\).
The product order (or block order) \(<\) \(:=\) \((<_1,<_2)\) on \(k[x,y]\) is defined as: \(x^a y^b < x^A y^B\) if and only if \(x^a <_1 x^A\) or (\(x^a =x^A\) and \(y^b <_2 y^B\)).
These block orders are constructed in Sage by giving a comma separated list of monomial orders with the length of each block attached to them.
EXAMPLES:
As an example, consider constructing a block order where the first four variables are compared using the degree reverse lexicographical order while the last two variables in the second block are compared using negative lexicographical order.
sage: P.<a,b,c,d,e,f> = PolynomialRing(QQ, 6, order='degrevlex(4),neglex(2)')
sage: a > c^4
False
sage: a > e^4
True
sage: e > f^2
False
>>> from sage.all import *
>>> P = PolynomialRing(QQ, Integer(6), order='degrevlex(4),neglex(2)', names=('a', 'b', 'c', 'd', 'e', 'f',)); (a, b, c, d, e, f,) = P._first_ngens(6)
>>> a > c**Integer(4)
False
>>> a > e**Integer(4)
True
>>> e > f**Integer(2)
False
The same result can be achieved by:
sage: T1 = TermOrder('degrevlex',4)
sage: T2 = TermOrder('neglex',2)
sage: T = T1 + T2
sage: P.<a,b,c,d,e,f> = PolynomialRing(QQ, 6, order=T)
sage: a > c^4
False
sage: a > e^4
True
>>> from sage.all import *
>>> T1 = TermOrder('degrevlex',Integer(4))
>>> T2 = TermOrder('neglex',Integer(2))
>>> T = T1 + T2
>>> P = PolynomialRing(QQ, Integer(6), order=T, names=('a', 'b', 'c', 'd', 'e', 'f',)); (a, b, c, d, e, f,) = P._first_ngens(6)
>>> a > c**Integer(4)
False
>>> a > e**Integer(4)
True
If any other unsupported term order is given the provided string can be forced to be passed through as is to Singular, Macaulay2, and Magma. This ensures that it is for example possible to calculate a Groebner basis with respect to some term order Singular supports but Sage doesn’t:
sage: T = TermOrder("royalorder")
Traceback (most recent call last):
...
ValueError: unknown term order 'royalorder'
sage: T = TermOrder("royalorder", force=True)
sage: T
royalorder term order
sage: T.singular_str()
'royalorder'
>>> from sage.all import *
>>> T = TermOrder("royalorder")
Traceback (most recent call last):
...
ValueError: unknown term order 'royalorder'
>>> T = TermOrder("royalorder", force=True)
>>> T
royalorder term order
>>> T.singular_str()
'royalorder'
AUTHORS:
David Joyner and William Stein: initial version of multi_polynomial_ring
Kiran S. Kedlaya: added macaulay2 interface
Martin Albrecht: implemented native term orders, refactoring
Kwankyu Lee: implemented matrix and weighted degree term orders
Simon King (2011-06-06): added termorder_from_singular
- class sage.rings.polynomial.term_order.TermOrder(name='lex', n=0, force=False)[source]¶
Bases:
SageObjectA term order.
See
sage.rings.polynomial.term_orderfor details on supported term orders.- blocks()[source]¶
Return the term order blocks of
self.NOTE:
This method has been added in Issue #11316. There used to be an attribute of the same name and the same content. So, it is a backward incompatible syntax change.
EXAMPLES:
sage: t = TermOrder('deglex',2) + TermOrder('lex',2) sage: t.blocks() (Degree lexicographic term order, Lexicographic term order)
>>> from sage.all import * >>> t = TermOrder('deglex',Integer(2)) + TermOrder('lex',Integer(2)) >>> t.blocks() (Degree lexicographic term order, Lexicographic term order)
- property greater_tuple¶
The default
greater_tuplemethod for this term order.EXAMPLES:
sage: O = TermOrder() sage: O.greater_tuple.__func__ is O.greater_tuple_lex.__func__ True sage: O = TermOrder('deglex') sage: O.greater_tuple.__func__ is O.greater_tuple_deglex.__func__ True
>>> from sage.all import * >>> O = TermOrder() >>> O.greater_tuple.__func__ is O.greater_tuple_lex.__func__ True >>> O = TermOrder('deglex') >>> O.greater_tuple.__func__ is O.greater_tuple_deglex.__func__ True
- greater_tuple_block(f, g)[source]¶
Return the greater exponent tuple with respect to the block order as specified when constructing this element.
This method is called by the lm/lc/lt methods of
MPolynomial_polydict.INPUT:
f– exponent tupleg– exponent tuple
EXAMPLES:
sage: P.<a,b,c,d,e,f> = PolynomialRing(QQbar, 6, # needs sage.rings.number_field ....: order='degrevlex(3),degrevlex(3)') sage: f = a + c^4; f.lm() # indirect doctest # needs sage.rings.number_field c^4 sage: g = a + e^4; g.lm() # needs sage.rings.number_field a
>>> from sage.all import * >>> P = PolynomialRing(QQbar, Integer(6), # needs sage.rings.number_field ... order='degrevlex(3),degrevlex(3)', names=('a', 'b', 'c', 'd', 'e', 'f',)); (a, b, c, d, e, f,) = P._first_ngens(6) >>> f = a + c**Integer(4); f.lm() # indirect doctest # needs sage.rings.number_field c^4 >>> g = a + e**Integer(4); g.lm() # needs sage.rings.number_field a
- greater_tuple_deglex(f, g)[source]¶
Return the greater exponent tuple with respect to the total degree lexicographical term order.
INPUT:
f– exponent tupleg– exponent tuple
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQbar, 3, order='deglex') # needs sage.rings.number_field sage: f = x + y; f.lm() # indirect doctest # needs sage.rings.number_field x sage: f = x + y^2*z; f.lm() # needs sage.rings.number_field y^2*z
>>> from sage.all import * >>> P = PolynomialRing(QQbar, Integer(3), order='deglex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)# needs sage.rings.number_field >>> f = x + y; f.lm() # indirect doctest # needs sage.rings.number_field x >>> f = x + y**Integer(2)*z; f.lm() # needs sage.rings.number_field y^2*z
This method is called by the lm/lc/lt methods of
MPolynomial_polydict.
- greater_tuple_degneglex(f, g)[source]¶
Return the greater exponent tuple with respect to the degree negative lexicographical term order.
INPUT:
f– exponent tupleg– exponent tuple
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQbar, 3, order='degneglex') # needs sage.rings.number_field sage: f = x + y; f.lm() # indirect doctest # needs sage.rings.number_field y sage: f = x + y^2*z; f.lm() # needs sage.rings.number_field y^2*z
>>> from sage.all import * >>> P = PolynomialRing(QQbar, Integer(3), order='degneglex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)# needs sage.rings.number_field >>> f = x + y; f.lm() # indirect doctest # needs sage.rings.number_field y >>> f = x + y**Integer(2)*z; f.lm() # needs sage.rings.number_field y^2*z
This method is called by the lm/lc/lt methods of
MPolynomial_polydict.
- greater_tuple_degrevlex(f, g)[source]¶
Return the greater exponent tuple with respect to the total degree reversed lexicographical term order.
INPUT:
f– exponent tupleg– exponent tuple
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQbar, 3, order='degrevlex') # needs sage.rings.number_field sage: f = x + y; f.lm() # indirect doctest # needs sage.rings.number_field x sage: f = x + y^2*z; f.lm() # needs sage.rings.number_field y^2*z
>>> from sage.all import * >>> P = PolynomialRing(QQbar, Integer(3), order='degrevlex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)# needs sage.rings.number_field >>> f = x + y; f.lm() # indirect doctest # needs sage.rings.number_field x >>> f = x + y**Integer(2)*z; f.lm() # needs sage.rings.number_field y^2*z
This method is called by the lm/lc/lt methods of
MPolynomial_polydict.
- greater_tuple_invlex(f, g)[source]¶
Return the greater exponent tuple with respect to the inversed lexicographical term order.
INPUT:
f– exponent tupleg– exponent tuple
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQbar, 3, order='invlex') # needs sage.rings.number_field sage: f = x + y; f.lm() # indirect doctest # needs sage.rings.number_field y sage: f = y + x^2; f.lm() # needs sage.rings.number_field y
>>> from sage.all import * >>> P = PolynomialRing(QQbar, Integer(3), order='invlex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)# needs sage.rings.number_field >>> f = x + y; f.lm() # indirect doctest # needs sage.rings.number_field y >>> f = y + x**Integer(2); f.lm() # needs sage.rings.number_field y
This method is called by the lm/lc/lt methods of
MPolynomial_polydict.
- greater_tuple_lex(f, g)[source]¶
Return the greater exponent tuple with respect to the lexicographical term order.
INPUT:
f– exponent tupleg– exponent tuple
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQbar, 3, order='lex') # needs sage.rings.number_field sage: f = x + y^2; f.lm() # indirect doctest # needs sage.rings.number_field x
>>> from sage.all import * >>> P = PolynomialRing(QQbar, Integer(3), order='lex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)# needs sage.rings.number_field >>> f = x + y**Integer(2); f.lm() # indirect doctest # needs sage.rings.number_field x
This method is called by the lm/lc/lt methods of
MPolynomial_polydict.
- greater_tuple_matrix(f, g)[source]¶
Return the greater exponent tuple with respect to the matrix term order.
INPUT:
f– exponent tupleg– exponent tuple
EXAMPLES:
sage: P.<x,y> = PolynomialRing(QQbar, 2, order='m(1,3,1,0)') # needs sage.rings.number_field sage: y > x^2 # indirect doctest # needs sage.rings.number_field True sage: y > x^3 # needs sage.rings.number_field False
>>> from sage.all import * >>> P = PolynomialRing(QQbar, Integer(2), order='m(1,3,1,0)', names=('x', 'y',)); (x, y,) = P._first_ngens(2)# needs sage.rings.number_field >>> y > x**Integer(2) # indirect doctest # needs sage.rings.number_field True >>> y > x**Integer(3) # needs sage.rings.number_field False
- greater_tuple_negdeglex(f, g)[source]¶
Return the greater exponent tuple with respect to the negative degree lexicographical term order.
INPUT:
f– exponent tupleg– exponent tuple
EXAMPLES:
sage: # needs sage.rings.number_field sage: P.<x,y,z> = PolynomialRing(QQbar, 3, order='negdeglex') sage: f = x + y; f.lm() # indirect doctest x sage: f = x + x^2; f.lm() x sage: f = x^2*y*z^2 + x*y^3*z; f.lm() x^2*y*z^2
>>> from sage.all import * >>> # needs sage.rings.number_field >>> P = PolynomialRing(QQbar, Integer(3), order='negdeglex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> f = x + y; f.lm() # indirect doctest x >>> f = x + x**Integer(2); f.lm() x >>> f = x**Integer(2)*y*z**Integer(2) + x*y**Integer(3)*z; f.lm() x^2*y*z^2
This method is called by the lm/lc/lt methods of
MPolynomial_polydict.
- greater_tuple_negdegrevlex(f, g)[source]¶
Return the greater exponent tuple with respect to the negative degree reverse lexicographical term order.
INPUT:
f– exponent tupleg– exponent tuple
EXAMPLES:
sage: # needs sage.rings.number_field sage: P.<x,y,z> = PolynomialRing(QQbar, 3, order='negdegrevlex') sage: f = x + y; f.lm() # indirect doctest x sage: f = x + x^2; f.lm() x sage: f = x^2*y*z^2 + x*y^3*z; f.lm() x*y^3*z
>>> from sage.all import * >>> # needs sage.rings.number_field >>> P = PolynomialRing(QQbar, Integer(3), order='negdegrevlex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> f = x + y; f.lm() # indirect doctest x >>> f = x + x**Integer(2); f.lm() x >>> f = x**Integer(2)*y*z**Integer(2) + x*y**Integer(3)*z; f.lm() x*y^3*z
This method is called by the lm/lc/lt methods of
MPolynomial_polydict.
- greater_tuple_neglex(f, g)[source]¶
Return the greater exponent tuple with respect to the negative lexicographical term order.
This method is called by the lm/lc/lt methods of
MPolynomial_polydict.INPUT:
f– exponent tupleg– exponent tuple
EXAMPLES:
sage: P.<a,b,c,d,e,f> = PolynomialRing(QQbar, 6, # needs sage.rings.number_field ....: order='degrevlex(3),degrevlex(3)') sage: f = a + c^4; f.lm() # indirect doctest # needs sage.rings.number_field c^4 sage: g = a + e^4; g.lm() # needs sage.rings.number_field a
>>> from sage.all import * >>> P = PolynomialRing(QQbar, Integer(6), # needs sage.rings.number_field ... order='degrevlex(3),degrevlex(3)', names=('a', 'b', 'c', 'd', 'e', 'f',)); (a, b, c, d, e, f,) = P._first_ngens(6) >>> f = a + c**Integer(4); f.lm() # indirect doctest # needs sage.rings.number_field c^4 >>> g = a + e**Integer(4); g.lm() # needs sage.rings.number_field a
- greater_tuple_negwdeglex(f, g)[source]¶
Return the greater exponent tuple with respect to the negative weighted degree lexicographical term order.
INPUT:
f– exponent tupleg– exponent tuple
EXAMPLES:
sage: # needs sage.rings.number_field sage: t = TermOrder('negwdeglex',(1,2,3)) sage: P.<x,y,z> = PolynomialRing(QQbar, 3, order=t) sage: f = x + y; f.lm() # indirect doctest x sage: f = x + x^2; f.lm() x sage: f = x^3 + z; f.lm() x^3
>>> from sage.all import * >>> # needs sage.rings.number_field >>> t = TermOrder('negwdeglex',(Integer(1),Integer(2),Integer(3))) >>> P = PolynomialRing(QQbar, Integer(3), order=t, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> f = x + y; f.lm() # indirect doctest x >>> f = x + x**Integer(2); f.lm() x >>> f = x**Integer(3) + z; f.lm() x^3
This method is called by the lm/lc/lt methods of
MPolynomial_polydict.
- greater_tuple_negwdegrevlex(f, g)[source]¶
Return the greater exponent tuple with respect to the negative weighted degree reverse lexicographical term order.
INPUT:
f– exponent tupleg– exponent tuple
EXAMPLES:
sage: # needs sage.rings.number_field sage: t = TermOrder('negwdegrevlex',(1,2,3)) sage: P.<x,y,z> = PolynomialRing(QQbar, 3, order=t) sage: f = x + y; f.lm() # indirect doctest x sage: f = x + x^2; f.lm() x sage: f = x^3 + z; f.lm() x^3
>>> from sage.all import * >>> # needs sage.rings.number_field >>> t = TermOrder('negwdegrevlex',(Integer(1),Integer(2),Integer(3))) >>> P = PolynomialRing(QQbar, Integer(3), order=t, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> f = x + y; f.lm() # indirect doctest x >>> f = x + x**Integer(2); f.lm() x >>> f = x**Integer(3) + z; f.lm() x^3
This method is called by the lm/lc/lt methods of
MPolynomial_polydict.
- greater_tuple_wdeglex(f, g)[source]¶
Return the greater exponent tuple with respect to the weighted degree lexicographical term order.
INPUT:
f– exponent tupleg– exponent tuple
EXAMPLES:
sage: t = TermOrder('wdeglex',(1,2,3)) sage: P.<x,y,z> = PolynomialRing(QQbar, 3, order=t) # needs sage.rings.number_field sage: f = x + y; f.lm() # indirect doctest # needs sage.rings.number_field y sage: f = x*y + z; f.lm() # needs sage.rings.number_field x*y
>>> from sage.all import * >>> t = TermOrder('wdeglex',(Integer(1),Integer(2),Integer(3))) >>> P = PolynomialRing(QQbar, Integer(3), order=t, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)# needs sage.rings.number_field >>> f = x + y; f.lm() # indirect doctest # needs sage.rings.number_field y >>> f = x*y + z; f.lm() # needs sage.rings.number_field x*y
This method is called by the lm/lc/lt methods of
MPolynomial_polydict.
- greater_tuple_wdegrevlex(f, g)[source]¶
Return the greater exponent tuple with respect to the weighted degree reverse lexicographical term order.
INPUT:
f– exponent tupleg– exponent tuple
EXAMPLES:
sage: t = TermOrder('wdegrevlex',(1,2,3)) sage: P.<x,y,z> = PolynomialRing(QQbar, 3, order=t) # needs sage.rings.number_field sage: f = x + y; f.lm() # indirect doctest # needs sage.rings.number_field y sage: f = x + y^2*z; f.lm() # needs sage.rings.number_field y^2*z
>>> from sage.all import * >>> t = TermOrder('wdegrevlex',(Integer(1),Integer(2),Integer(3))) >>> P = PolynomialRing(QQbar, Integer(3), order=t, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)# needs sage.rings.number_field >>> f = x + y; f.lm() # indirect doctest # needs sage.rings.number_field y >>> f = x + y**Integer(2)*z; f.lm() # needs sage.rings.number_field y^2*z
This method is called by the lm/lc/lt methods of
MPolynomial_polydict.
- is_block_order()[source]¶
Return
Trueifselfis a block term order.EXAMPLES:
sage: t = TermOrder('deglex',2) + TermOrder('lex',2) sage: t.is_block_order() True
>>> from sage.all import * >>> t = TermOrder('deglex',Integer(2)) + TermOrder('lex',Integer(2)) >>> t.is_block_order() True
- is_global()[source]¶
Return
Trueif this term order is definitely global. Return false otherwise, which includes unknown term orders.EXAMPLES:
sage: T = TermOrder('lex') sage: T.is_global() True sage: T = TermOrder('degrevlex', 3) + TermOrder('degrevlex', 3) sage: T.is_global() True sage: T = TermOrder('degrevlex', 3) + TermOrder('negdegrevlex', 3) sage: T.is_global() False sage: T = TermOrder('degneglex', 3) sage: T.is_global() True sage: T = TermOrder('invlex', 3) sage: T.is_global() True
>>> from sage.all import * >>> T = TermOrder('lex') >>> T.is_global() True >>> T = TermOrder('degrevlex', Integer(3)) + TermOrder('degrevlex', Integer(3)) >>> T.is_global() True >>> T = TermOrder('degrevlex', Integer(3)) + TermOrder('negdegrevlex', Integer(3)) >>> T.is_global() False >>> T = TermOrder('degneglex', Integer(3)) >>> T.is_global() True >>> T = TermOrder('invlex', Integer(3)) >>> T.is_global() True
- is_local()[source]¶
Return
Trueif this term order is definitely local. Return false otherwise, which includes unknown term orders.EXAMPLES:
sage: T = TermOrder('lex') sage: T.is_local() False sage: T = TermOrder('negdeglex', 3) + TermOrder('negdegrevlex', 3) sage: T.is_local() True sage: T = TermOrder('degrevlex', 3) + TermOrder('negdegrevlex', 3) sage: T.is_local() False
>>> from sage.all import * >>> T = TermOrder('lex') >>> T.is_local() False >>> T = TermOrder('negdeglex', Integer(3)) + TermOrder('negdegrevlex', Integer(3)) >>> T.is_local() True >>> T = TermOrder('degrevlex', Integer(3)) + TermOrder('negdegrevlex', Integer(3)) >>> T.is_local() False
- is_weighted_degree_order()[source]¶
Return
Trueifselfis a weighted degree term order.EXAMPLES:
sage: t = TermOrder('wdeglex',(2,3)) sage: t.is_weighted_degree_order() True
>>> from sage.all import * >>> t = TermOrder('wdeglex',(Integer(2),Integer(3))) >>> t.is_weighted_degree_order() True
- macaulay2_str()[source]¶
Return a Macaulay2 representation of
self.Used to convert polynomial rings to their Macaulay2 representation.
EXAMPLES:
sage: P = PolynomialRing(GF(127), 8, names='x', order='degrevlex(3),lex(5)') sage: T = P.term_order() sage: T.macaulay2_str() '{GRevLex => 3,Lex => 5}' sage: P._macaulay2_().options()['MonomialOrder'] # optional - macaulay2 {MonomialSize => 16 } {GRevLex => {1, 1, 1}} {Lex => 5 } {Position => Up }
>>> from sage.all import * >>> P = PolynomialRing(GF(Integer(127)), Integer(8), names='x', order='degrevlex(3),lex(5)') >>> T = P.term_order() >>> T.macaulay2_str() '{GRevLex => 3,Lex => 5}' >>> P._macaulay2_().options()['MonomialOrder'] # optional - macaulay2 {MonomialSize => 16 } {GRevLex => {1, 1, 1}} {Lex => 5 } {Position => Up }
- magma_str()[source]¶
Return a MAGMA representation of
self.Used to convert polynomial rings to their MAGMA representation.
EXAMPLES:
sage: P = PolynomialRing(GF(127), 10, names='x', order='degrevlex') sage: magma(P) # optional - magma Polynomial ring of rank 10 over GF(127) Order: Graded Reverse Lexicographical Variables: x0, x1, x2, x3, x4, x5, x6, x7, x8, x9
>>> from sage.all import * >>> P = PolynomialRing(GF(Integer(127)), Integer(10), names='x', order='degrevlex') >>> magma(P) # optional - magma Polynomial ring of rank 10 over GF(127) Order: Graded Reverse Lexicographical Variables: x0, x1, x2, x3, x4, x5, x6, x7, x8, x9
sage: T = P.term_order() sage: T.magma_str() '"grevlex"'
>>> from sage.all import * >>> T = P.term_order() >>> T.magma_str() '"grevlex"'
- matrix()[source]¶
Return the matrix defining matrix term order.
EXAMPLES:
sage: t = TermOrder("M(1,2,0,1)") # needs sage.modules sage: t.matrix() # needs sage.modules [1 2] [0 1]
>>> from sage.all import * >>> t = TermOrder("M(1,2,0,1)") # needs sage.modules >>> t.matrix() # needs sage.modules [1 2] [0 1]
- name()[source]¶
EXAMPLES:
sage: TermOrder('lex').name() 'lex'
>>> from sage.all import * >>> TermOrder('lex').name() 'lex'
- singular_moreblocks()[source]¶
Return a the number of additional blocks SINGULAR needs to allocate for handling non-native orderings like
degneglex.EXAMPLES:
sage: P = PolynomialRing(GF(127), 10, names='x', ....: order='lex(3),deglex(5),lex(2)') sage: T = P.term_order() sage: T.singular_moreblocks() 0 sage: P = PolynomialRing(GF(127), 10, names='x', ....: order='lex(3),degneglex(5),lex(2)') sage: T = P.term_order() sage: T.singular_moreblocks() 1 sage: P = PolynomialRing(GF(127), 10, names='x', ....: order='degneglex(5),degneglex(5)') sage: T = P.term_order() sage: T.singular_moreblocks() 2
>>> from sage.all import * >>> P = PolynomialRing(GF(Integer(127)), Integer(10), names='x', ... order='lex(3),deglex(5),lex(2)') >>> T = P.term_order() >>> T.singular_moreblocks() 0 >>> P = PolynomialRing(GF(Integer(127)), Integer(10), names='x', ... order='lex(3),degneglex(5),lex(2)') >>> T = P.term_order() >>> T.singular_moreblocks() 1 >>> P = PolynomialRing(GF(Integer(127)), Integer(10), names='x', ... order='degneglex(5),degneglex(5)') >>> T = P.term_order() >>> T.singular_moreblocks() 2
- singular_str()[source]¶
Return a SINGULAR representation of
self.Used to convert polynomial rings to their SINGULAR representation.
EXAMPLES:
sage: P = PolynomialRing(GF(127), 10, names='x', ....: order='lex(3),deglex(5),lex(2)') sage: T = P.term_order() sage: T.singular_str() '(lp(3),Dp(5),lp(2))' sage: P._singular_() # needs sage.libs.singular polynomial ring, over a field, global ordering // coefficients: ZZ/127 // number of vars : 10 // block 1 : ordering lp // : names x0 x1 x2 // block 2 : ordering Dp // : names x3 x4 x5 x6 x7 // block 3 : ordering lp // : names x8 x9 // block 4 : ordering C
>>> from sage.all import * >>> P = PolynomialRing(GF(Integer(127)), Integer(10), names='x', ... order='lex(3),deglex(5),lex(2)') >>> T = P.term_order() >>> T.singular_str() '(lp(3),Dp(5),lp(2))' >>> P._singular_() # needs sage.libs.singular polynomial ring, over a field, global ordering // coefficients: ZZ/127 // number of vars : 10 // block 1 : ordering lp // : names x0 x1 x2 // block 2 : ordering Dp // : names x3 x4 x5 x6 x7 // block 3 : ordering lp // : names x8 x9 // block 4 : ordering C
The
degneglexordering is somehow special, it looks like a block ordering in SINGULAR:sage: T = TermOrder("degneglex", 2) sage: P = PolynomialRing(QQ,2, names='x', order=T) sage: T = P.term_order() sage: T.singular_str() '(a(1:2),ls(2))' sage: T = TermOrder("degneglex", 2) + TermOrder("degneglex", 2) sage: P = PolynomialRing(QQ,4, names='x', order=T) sage: T = P.term_order() sage: T.singular_str() '(a(1:2),ls(2),a(1:2),ls(2))' sage: P._singular_() # needs sage.libs.singular polynomial ring, over a field, global ordering // coefficients: QQ // number of vars : 4 // block 1 : ordering a // : names x0 x1 // : weights 1 1 // block 2 : ordering ls // : names x0 x1 // block 3 : ordering a // : names x2 x3 // : weights 1 1 // block 4 : ordering ls // : names x2 x3 // block 5 : ordering C
>>> from sage.all import * >>> T = TermOrder("degneglex", Integer(2)) >>> P = PolynomialRing(QQ,Integer(2), names='x', order=T) >>> T = P.term_order() >>> T.singular_str() '(a(1:2),ls(2))' >>> T = TermOrder("degneglex", Integer(2)) + TermOrder("degneglex", Integer(2)) >>> P = PolynomialRing(QQ,Integer(4), names='x', order=T) >>> T = P.term_order() >>> T.singular_str() '(a(1:2),ls(2),a(1:2),ls(2))' >>> P._singular_() # needs sage.libs.singular polynomial ring, over a field, global ordering // coefficients: QQ // number of vars : 4 // block 1 : ordering a // : names x0 x1 // : weights 1 1 // block 2 : ordering ls // : names x0 x1 // block 3 : ordering a // : names x2 x3 // : weights 1 1 // block 4 : ordering ls // : names x2 x3 // block 5 : ordering C
The position of the
ordering Cblock can be controlled by setting_singular_ringorder_columnattribute to an integer:sage: T = TermOrder("degneglex", 2) + TermOrder("degneglex", 2) sage: T._singular_ringorder_column = 0 sage: P = PolynomialRing(QQ, 4, names='x', order=T) sage: P._singular_() # needs sage.libs.singular polynomial ring, over a field, global ordering // coefficients: QQ // number of vars : 4 // block 1 : ordering C // block 2 : ordering a // : names x0 x1 // : weights 1 1 // block 3 : ordering ls // : names x0 x1 // block 4 : ordering a // : names x2 x3 // : weights 1 1 // block 5 : ordering ls // : names x2 x3 sage: T._singular_ringorder_column = 1 sage: P = PolynomialRing(QQ, 4, names='y', order=T) sage: P._singular_() # needs sage.libs.singular polynomial ring, over a field, global ordering // coefficients: QQ // number of vars : 4 // block 1 : ordering c // block 2 : ordering a // : names y0 y1 // : weights 1 1 // block 3 : ordering ls // : names y0 y1 // block 4 : ordering a // : names y2 y3 // : weights 1 1 // block 5 : ordering ls // : names y2 y3 sage: T._singular_ringorder_column = 2 sage: P = PolynomialRing(QQ, 4, names='z', order=T) sage: P._singular_() # needs sage.libs.singular polynomial ring, over a field, global ordering // coefficients: QQ // number of vars : 4 // block 1 : ordering a // : names z0 z1 // : weights 1 1 // block 2 : ordering C // block 3 : ordering ls // : names z0 z1 // block 4 : ordering a // : names z2 z3 // : weights 1 1 // block 5 : ordering ls // : names z2 z3
>>> from sage.all import * >>> T = TermOrder("degneglex", Integer(2)) + TermOrder("degneglex", Integer(2)) >>> T._singular_ringorder_column = Integer(0) >>> P = PolynomialRing(QQ, Integer(4), names='x', order=T) >>> P._singular_() # needs sage.libs.singular polynomial ring, over a field, global ordering // coefficients: QQ // number of vars : 4 // block 1 : ordering C // block 2 : ordering a // : names x0 x1 // : weights 1 1 // block 3 : ordering ls // : names x0 x1 // block 4 : ordering a // : names x2 x3 // : weights 1 1 // block 5 : ordering ls // : names x2 x3 >>> T._singular_ringorder_column = Integer(1) >>> P = PolynomialRing(QQ, Integer(4), names='y', order=T) >>> P._singular_() # needs sage.libs.singular polynomial ring, over a field, global ordering // coefficients: QQ // number of vars : 4 // block 1 : ordering c // block 2 : ordering a // : names y0 y1 // : weights 1 1 // block 3 : ordering ls // : names y0 y1 // block 4 : ordering a // : names y2 y3 // : weights 1 1 // block 5 : ordering ls // : names y2 y3 >>> T._singular_ringorder_column = Integer(2) >>> P = PolynomialRing(QQ, Integer(4), names='z', order=T) >>> P._singular_() # needs sage.libs.singular polynomial ring, over a field, global ordering // coefficients: QQ // number of vars : 4 // block 1 : ordering a // : names z0 z1 // : weights 1 1 // block 2 : ordering C // block 3 : ordering ls // : names z0 z1 // block 4 : ordering a // : names z2 z3 // : weights 1 1 // block 5 : ordering ls // : names z2 z3
- property sortkey¶
The default
sortkeymethod for this term order.EXAMPLES:
sage: O = TermOrder() sage: O.sortkey.__func__ is O.sortkey_lex.__func__ True sage: O = TermOrder('deglex') sage: O.sortkey.__func__ is O.sortkey_deglex.__func__ True
>>> from sage.all import * >>> O = TermOrder() >>> O.sortkey.__func__ is O.sortkey_lex.__func__ True >>> O = TermOrder('deglex') >>> O.sortkey.__func__ is O.sortkey_deglex.__func__ True
- sortkey_block(f)[source]¶
Return the sortkey of an exponent tuple with respect to the block order as specified when constructing this element.
INPUT:
f– exponent tuple
EXAMPLES:
sage: P.<a,b,c,d,e,f> = PolynomialRing(QQbar, 6, # needs sage.rings.number_field ....: order='degrevlex(3),degrevlex(3)') sage: a > c^4 # indirect doctest # needs sage.rings.number_field False sage: a > e^4 # needs sage.rings.number_field True
>>> from sage.all import * >>> P = PolynomialRing(QQbar, Integer(6), # needs sage.rings.number_field ... order='degrevlex(3),degrevlex(3)', names=('a', 'b', 'c', 'd', 'e', 'f',)); (a, b, c, d, e, f,) = P._first_ngens(6) >>> a > c**Integer(4) # indirect doctest # needs sage.rings.number_field False >>> a > e**Integer(4) # needs sage.rings.number_field True
- sortkey_deglex(f)[source]¶
Return the sortkey of an exponent tuple with respect to the degree lexicographical term order.
INPUT:
f– exponent tuple
EXAMPLES:
sage: P.<x,y> = PolynomialRing(QQbar, 2, order='deglex') # needs sage.rings.number_field sage: x > y^2 # indirect doctest # needs sage.rings.number_field False sage: x > 1 # needs sage.rings.number_field True
>>> from sage.all import * >>> P = PolynomialRing(QQbar, Integer(2), order='deglex', names=('x', 'y',)); (x, y,) = P._first_ngens(2)# needs sage.rings.number_field >>> x > y**Integer(2) # indirect doctest # needs sage.rings.number_field False >>> x > Integer(1) # needs sage.rings.number_field True
- sortkey_degneglex(f)[source]¶
Return the sortkey of an exponent tuple with respect to the degree negative lexicographical term order.
INPUT:
f– exponent tuple
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQbar, 3, order='degneglex') # needs sage.rings.number_field sage: x*y > y*z # indirect doctest # needs sage.rings.number_field False sage: x*y > x # needs sage.rings.number_field True
>>> from sage.all import * >>> P = PolynomialRing(QQbar, Integer(3), order='degneglex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)# needs sage.rings.number_field >>> x*y > y*z # indirect doctest # needs sage.rings.number_field False >>> x*y > x # needs sage.rings.number_field True
- sortkey_degrevlex(f)[source]¶
Return the sortkey of an exponent tuple with respect to the degree reversed lexicographical term order.
INPUT:
f– exponent tuple
EXAMPLES:
sage: P.<x,y> = PolynomialRing(QQbar, 2, order='degrevlex') # needs sage.rings.number_field sage: x > y^2 # indirect doctest # needs sage.rings.number_field False sage: x > 1 # needs sage.rings.number_field True
>>> from sage.all import * >>> P = PolynomialRing(QQbar, Integer(2), order='degrevlex', names=('x', 'y',)); (x, y,) = P._first_ngens(2)# needs sage.rings.number_field >>> x > y**Integer(2) # indirect doctest # needs sage.rings.number_field False >>> x > Integer(1) # needs sage.rings.number_field True
- sortkey_invlex(f)[source]¶
Return the sortkey of an exponent tuple with respect to the inversed lexicographical term order.
INPUT:
f– exponent tuple
EXAMPLES:
sage: P.<x,y> = PolynomialRing(QQbar, 2, order='invlex') # needs sage.rings.number_field sage: x > y^2 # indirect doctest # needs sage.rings.number_field False sage: x > 1 # needs sage.rings.number_field True
>>> from sage.all import * >>> P = PolynomialRing(QQbar, Integer(2), order='invlex', names=('x', 'y',)); (x, y,) = P._first_ngens(2)# needs sage.rings.number_field >>> x > y**Integer(2) # indirect doctest # needs sage.rings.number_field False >>> x > Integer(1) # needs sage.rings.number_field True
- sortkey_lex(f)[source]¶
Return the sortkey of an exponent tuple with respect to the lexicographical term order.
INPUT:
f– exponent tuple
EXAMPLES:
sage: P.<x,y> = PolynomialRing(QQbar, 2, order='lex') # needs sage.rings.number_field sage: x > y^2 # indirect doctest # needs sage.rings.number_field True sage: x > 1 # needs sage.rings.number_field True
>>> from sage.all import * >>> P = PolynomialRing(QQbar, Integer(2), order='lex', names=('x', 'y',)); (x, y,) = P._first_ngens(2)# needs sage.rings.number_field >>> x > y**Integer(2) # indirect doctest # needs sage.rings.number_field True >>> x > Integer(1) # needs sage.rings.number_field True
- sortkey_matrix(f)[source]¶
Return the sortkey of an exponent tuple with respect to the matrix term order.
INPUT:
f– exponent tuple
EXAMPLES:
sage: P.<x,y> = PolynomialRing(QQbar, 2, order='m(1,3,1,0)') # needs sage.rings.number_field sage: y > x^2 # indirect doctest # needs sage.rings.number_field True sage: y > x^3 # needs sage.rings.number_field False
>>> from sage.all import * >>> P = PolynomialRing(QQbar, Integer(2), order='m(1,3,1,0)', names=('x', 'y',)); (x, y,) = P._first_ngens(2)# needs sage.rings.number_field >>> y > x**Integer(2) # indirect doctest # needs sage.rings.number_field True >>> y > x**Integer(3) # needs sage.rings.number_field False
- sortkey_negdeglex(f)[source]¶
Return the sortkey of an exponent tuple with respect to the negative degree lexicographical term order.
INPUT:
f– exponent tuple
EXAMPLES:
sage: P.<x,y> = PolynomialRing(QQbar, 2, order='negdeglex') # needs sage.rings.number_field sage: x > y^2 # indirect doctest # needs sage.rings.number_field True sage: x > 1 # needs sage.rings.number_field False
>>> from sage.all import * >>> P = PolynomialRing(QQbar, Integer(2), order='negdeglex', names=('x', 'y',)); (x, y,) = P._first_ngens(2)# needs sage.rings.number_field >>> x > y**Integer(2) # indirect doctest # needs sage.rings.number_field True >>> x > Integer(1) # needs sage.rings.number_field False
- sortkey_negdegrevlex(f)[source]¶
Return the sortkey of an exponent tuple with respect to the negative degree reverse lexicographical term order.
INPUT:
f– exponent tuple
EXAMPLES:
sage: P.<x,y> = PolynomialRing(QQbar, 2, order='negdegrevlex') # needs sage.rings.number_field sage: x > y^2 # indirect doctest # needs sage.rings.number_field True sage: x > 1 # needs sage.rings.number_field False
>>> from sage.all import * >>> P = PolynomialRing(QQbar, Integer(2), order='negdegrevlex', names=('x', 'y',)); (x, y,) = P._first_ngens(2)# needs sage.rings.number_field >>> x > y**Integer(2) # indirect doctest # needs sage.rings.number_field True >>> x > Integer(1) # needs sage.rings.number_field False
- sortkey_neglex(f)[source]¶
Return the sortkey of an exponent tuple with respect to the negative lexicographical term order.
INPUT:
f– exponent tuple
EXAMPLES:
sage: P.<x,y> = PolynomialRing(QQbar, 2, order='neglex') # needs sage.rings.number_field sage: x > y^2 # indirect doctest # needs sage.rings.number_field False sage: x > 1 # needs sage.rings.number_field False
>>> from sage.all import * >>> P = PolynomialRing(QQbar, Integer(2), order='neglex', names=('x', 'y',)); (x, y,) = P._first_ngens(2)# needs sage.rings.number_field >>> x > y**Integer(2) # indirect doctest # needs sage.rings.number_field False >>> x > Integer(1) # needs sage.rings.number_field False
- sortkey_negwdeglex(f)[source]¶
Return the sortkey of an exponent tuple with respect to the negative weighted degree lexicographical term order.
INPUT:
f– exponent tuple
EXAMPLES:
sage: t = TermOrder('negwdeglex',(3,2)) sage: P.<x,y> = PolynomialRing(QQbar, 2, order=t) # needs sage.rings.number_field sage: x > y^2 # indirect doctest # needs sage.rings.number_field True sage: x^2 > y^3 # needs sage.rings.number_field True
>>> from sage.all import * >>> t = TermOrder('negwdeglex',(Integer(3),Integer(2))) >>> P = PolynomialRing(QQbar, Integer(2), order=t, names=('x', 'y',)); (x, y,) = P._first_ngens(2)# needs sage.rings.number_field >>> x > y**Integer(2) # indirect doctest # needs sage.rings.number_field True >>> x**Integer(2) > y**Integer(3) # needs sage.rings.number_field True
- sortkey_negwdegrevlex(f)[source]¶
Return the sortkey of an exponent tuple with respect to the negative weighted degree reverse lexicographical term order.
INPUT:
f– exponent tuple
EXAMPLES:
sage: t = TermOrder('negwdegrevlex',(3,2)) sage: P.<x,y> = PolynomialRing(QQbar, 2, order=t) # needs sage.rings.number_field sage: x > y^2 # indirect doctest # needs sage.rings.number_field True sage: x^2 > y^3 # needs sage.rings.number_field True
>>> from sage.all import * >>> t = TermOrder('negwdegrevlex',(Integer(3),Integer(2))) >>> P = PolynomialRing(QQbar, Integer(2), order=t, names=('x', 'y',)); (x, y,) = P._first_ngens(2)# needs sage.rings.number_field >>> x > y**Integer(2) # indirect doctest # needs sage.rings.number_field True >>> x**Integer(2) > y**Integer(3) # needs sage.rings.number_field True
- sortkey_wdeglex(f)[source]¶
Return the sortkey of an exponent tuple with respect to the weighted degree lexicographical term order.
INPUT:
f– exponent tuple
EXAMPLES:
sage: t = TermOrder('wdeglex',(3,2)) sage: P.<x,y> = PolynomialRing(QQbar, 2, order=t) # needs sage.rings.number_field sage: x > y^2 # indirect doctest # needs sage.rings.number_field False sage: x > y # needs sage.rings.number_field True
>>> from sage.all import * >>> t = TermOrder('wdeglex',(Integer(3),Integer(2))) >>> P = PolynomialRing(QQbar, Integer(2), order=t, names=('x', 'y',)); (x, y,) = P._first_ngens(2)# needs sage.rings.number_field >>> x > y**Integer(2) # indirect doctest # needs sage.rings.number_field False >>> x > y # needs sage.rings.number_field True
- sortkey_wdegrevlex(f)[source]¶
Return the sortkey of an exponent tuple with respect to the weighted degree reverse lexicographical term order.
INPUT:
f– exponent tuple
EXAMPLES:
sage: t = TermOrder('wdegrevlex',(3,2)) sage: P.<x,y> = PolynomialRing(QQbar, 2, order=t) # needs sage.rings.number_field sage: x > y^2 # indirect doctest # needs sage.rings.number_field False sage: x^2 > y^3 # needs sage.rings.number_field True
>>> from sage.all import * >>> t = TermOrder('wdegrevlex',(Integer(3),Integer(2))) >>> P = PolynomialRing(QQbar, Integer(2), order=t, names=('x', 'y',)); (x, y,) = P._first_ngens(2)# needs sage.rings.number_field >>> x > y**Integer(2) # indirect doctest # needs sage.rings.number_field False >>> x**Integer(2) > y**Integer(3) # needs sage.rings.number_field True
- tuple_weight(f)[source]¶
Return the weight of tuple f.
INPUT:
f– exponent tuple
EXAMPLES:
sage: t = TermOrder('wdeglex',(1,2,3)) sage: P.<a,b,c> = PolynomialRing(QQbar, order=t) # needs sage.rings.number_field sage: P.term_order().tuple_weight([3,2,1]) # needs sage.rings.number_field 10
>>> from sage.all import * >>> t = TermOrder('wdeglex',(Integer(1),Integer(2),Integer(3))) >>> P = PolynomialRing(QQbar, order=t, names=('a', 'b', 'c',)); (a, b, c,) = P._first_ngens(3)# needs sage.rings.number_field >>> P.term_order().tuple_weight([Integer(3),Integer(2),Integer(1)]) # needs sage.rings.number_field 10
- sage.rings.polynomial.term_order.termorder_from_singular(S)[source]¶
Return the Sage term order of the basering in the given Singular interface.
INPUT:
An instance of the Singular interface.
EXAMPLES:
sage: from sage.rings.polynomial.term_order import termorder_from_singular sage: singular.eval('ring r1 = (9,x),(a,b,c,d,e,f),(M((1,2,3,0)),wp(2,3),lp)') # needs sage.libs.singular '' sage: termorder_from_singular(singular) # needs sage.libs.singular Block term order with blocks: (Matrix term order with matrix [1 2] [3 0], Weighted degree reverse lexicographic term order with weights (2, 3), Lexicographic term order of length 2)
>>> from sage.all import * >>> from sage.rings.polynomial.term_order import termorder_from_singular >>> singular.eval('ring r1 = (9,x),(a,b,c,d,e,f),(M((1,2,3,0)),wp(2,3),lp)') # needs sage.libs.singular '' >>> termorder_from_singular(singular) # needs sage.libs.singular Block term order with blocks: (Matrix term order with matrix [1 2] [3 0], Weighted degree reverse lexicographic term order with weights (2, 3), Lexicographic term order of length 2)
A term order in Singular also involves information on orders for modules. This information is reflected in
_singular_ringorder_columnattribute of the term order.sage: # needs sage.libs.singular sage: singular.ring(0, '(x,y,z,w)', '(C,dp(2),lp(2))') polynomial ring, over a field, global ordering // coefficients: QQ // number of vars : 4 // block 1 : ordering C // block 2 : ordering dp // : names x y // block 3 : ordering lp // : names z w sage: T = termorder_from_singular(singular) sage: T Block term order with blocks: (Degree reverse lexicographic term order of length 2, Lexicographic term order of length 2) sage: T._singular_ringorder_column 0 sage: # needs sage.libs.singular sage: singular.ring(0, '(x,y,z,w)', '(c,dp(2),lp(2))') polynomial ring, over a field, global ordering // coefficients: QQ // number of vars : 4 // block 1 : ordering c // block 2 : ordering dp // : names x y // block 3 : ordering lp // : names z w sage: T = termorder_from_singular(singular) sage: T Block term order with blocks: (Degree reverse lexicographic term order of length 2, Lexicographic term order of length 2) sage: T._singular_ringorder_column 1
>>> from sage.all import * >>> # needs sage.libs.singular >>> singular.ring(Integer(0), '(x,y,z,w)', '(C,dp(2),lp(2))') polynomial ring, over a field, global ordering // coefficients: QQ // number of vars : 4 // block 1 : ordering C // block 2 : ordering dp // : names x y // block 3 : ordering lp // : names z w >>> T = termorder_from_singular(singular) >>> T Block term order with blocks: (Degree reverse lexicographic term order of length 2, Lexicographic term order of length 2) >>> T._singular_ringorder_column 0 >>> # needs sage.libs.singular >>> singular.ring(Integer(0), '(x,y,z,w)', '(c,dp(2),lp(2))') polynomial ring, over a field, global ordering // coefficients: QQ // number of vars : 4 // block 1 : ordering c // block 2 : ordering dp // : names x y // block 3 : ordering lp // : names z w >>> T = termorder_from_singular(singular) >>> T Block term order with blocks: (Degree reverse lexicographic term order of length 2, Lexicographic term order of length 2) >>> T._singular_ringorder_column 1