def test_build_product_order():
from sympy.abc import x, y, z, t
assert build_product_order((("grlex", x, y), ("grlex", z, t)), [x, y, z, t]) \
((4, 5, 6, 7)) == ((9, (4, 5)), (13, (6, 7)))
assert build_product_order((("grlex", x, y), ("grlex", z, t)), [x, y, z, t]) == \
build_product_order((("grlex", x, y), ("grlex", z, t)), [x, y, z, t])
assert (build_product_order((("grlex", x, y), ("grlex", z, t)), [x, y, z, t]) != \
build_product_order((("grlex", x, y), ("grlex", z, t)), [x, y, z, t])) \
is False
def test_build_product_order():
from sympy.abc import x, y, z, t
assert build_product_order((("grlex", x, y), ("grlex", z, t)), [x, y, z, t]) \
((4, 5, 6, 7)) == ((9, (4, 5)), (13, (6, 7)))
assert build_product_order((("grlex", x, y), ("grlex", z, t)), [x, y, z, t]) == \
build_product_order((("grlex", x, y), ("grlex", z, t)), [x, y, z, t])
assert (build_product_order((("grlex", x, y), ("grlex", z, t)), [x, y, z, t]) != \
build_product_order((("grlex", x, y), ("grlex", z, t)), [x, y, z, t])) \
is False
def PolynomialRing(dom, *gens, **opts):
r"""
Create a generalized multivariate polynomial ring.
A generalized polynomial ring is defined by a ground field `K`, a set
of generators (typically `x_1, \dots, x_n`) and a monomial order `<`.
The monomial order can be global, local or mixed. In any case it induces
a total ordering on the monomials, and there exists for every (non-zero)
polynomial `f \in K[x_1, \dots, x_n]` a well-defined "leading monomial"
`LM(f) = LM(f, >)`. One can then define a multiplicative subset
`S = S_> = \{f \in K[x_1, \dots, x_n] | LM(f) = 1\}`. The generalized
polynomial ring corresponding to the monomial order is
`R = S^{-1}K[x_1, \dots, x_n]`.
If `>` is a so-called global order, that is `1` is the smallest monomial,
then we just have `S = K` and `R = K[x_1, \dots, x_n]`.
Examples
========
A few examples may make this clearer.
>>> from sympy.abc import x, y
>>> from sympy import QQ
Our first ring uses global lexicographic order.
>>> R1 = QQ.poly_ring(x, y, order=(("lex", x, y),))
The second ring uses local lexicographic order. Note that when using a
single (non-product) order, you can just specify the name and omit the
variables:
>>> R2 = QQ.poly_ring(x, y, order="ilex")
The third and fourth rings use a mixed orders:
>>> o1 = (("ilex", x), ("lex", y))
>>> o2 = (("lex", x), ("ilex", y))
>>> R3 = QQ.poly_ring(x, y, order=o1)
>>> R4 = QQ.poly_ring(x, y, order=o2)
We will investigate what elements of `K(x, y)` are contained in the various
rings.
>>> L = [x, 1/x, y/(1 + x), 1/(1 + y), 1/(1 + x*y)]
>>> test = lambda R: [f in R for f in L]
The first ring is just `K[x, y]`:
>>> test(R1)
[True, False, False, False, False]
The second ring is R1 localised at the maximal ideal (x, y):
>>> test(R2)
[True, False, True, True, True]
The third ring is R1 localised at the prime ideal (x):
>>> test(R3)
[True, False, True, False, True]
Finally the fourth ring is R1 localised at `S = K[x, y] \setminus yK[y]`:
>>> test(R4)
[True, False, False, True, False]
"""
order = opts.get("order", GeneralizedPolynomialRing.default_order)
if iterable(order):
order = build_product_order(order, gens)
order = monomial_key(order)
opts['order'] = order
if order.is_global:
return GlobalPolynomialRing(dom, *gens, **opts)
else:
return GeneralizedPolynomialRing(dom, *gens, **opts)
def PolynomialRing(dom, *gens, **opts):
r"""
Create a generalized multivariate polynomial ring.
A generalized polynomial ring is defined by a ground field `K`, a set
of generators (typically `x_1, \dots, x_n`) and a monomial order `<`.
The monomial order can be global, local or mixed. In any case it induces
a total ordering on the monomials, and there exists for every (non-zero)
polynomial `f \in K[x_1, \dots, x_n]` a well-defined "leading monomial"
`LM(f) = LM(f, >)`. One can then define a multiplicative subset
`S = S_> = \{f \in K[x_1, \dots, x_n] | LM(f) = 1\}`. The generalized
polynomial ring corresponding to the monomial order is
`R = S^{-1}K[x_1, \dots, x_n]`.
If `>` is a so-called global order, that is `1` is the smallest monomial,
then we just have `S = K` and `R = K[x_1, \dots, x_n]`.
Examples
========
A few examples may make this clearer.
>>> from sympy.abc import x, y
>>> from sympy import QQ
Our first ring uses global lexicographic order.
>>> R1 = QQ.poly_ring(x, y, order=(("lex", x, y),))
The second ring uses local lexicographic order. Note that when using a
single (non-product) order, you can just specify the name and omit the
variables:
>>> R2 = QQ.poly_ring(x, y, order="ilex")
The third and fourth rings use a mixed orders:
>>> o1 = (("ilex", x), ("lex", y))
>>> o2 = (("lex", x), ("ilex", y))
>>> R3 = QQ.poly_ring(x, y, order=o1)
>>> R4 = QQ.poly_ring(x, y, order=o2)
We will investigate what elements of `K(x, y)` are contained in the various
rings.
>>> L = [x, 1/x, y/(1 + x), 1/(1 + y), 1/(1 + x*y)]
>>> test = lambda R: [f in R for f in L]
The first ring is just `K[x, y]`:
>>> test(R1)
[True, False, False, False, False]
The second ring is R1 localised at the maximal ideal (x, y):
>>> test(R2)
[True, False, True, True, True]
The third ring is R1 localised at the prime ideal (x):
>>> test(R3)
[True, False, True, False, True]
Finally the fourth ring is R1 localised at `S = K[x, y] \setminus yK[y]`:
>>> test(R4)
[True, False, False, True, False]
"""
order = opts.get("order", GeneralizedPolynomialRing.default_order)
if iterable(order):
order = build_product_order(order, gens)
order = monomial_key(order)
opts['order'] = order
if order.is_global:
return GlobalPolynomialRing(dom, *gens, **opts)
else:
return GeneralizedPolynomialRing(dom, *gens, **opts)
