def monomial_lcm(self, f, g):
"""
LCM for monomials. Coefficients are ignored.
INPUT:
- ``f`` - monomial
- ``g`` - monomial
EXAMPLE::
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain
sage: P.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex')
sage: P.monomial_lcm(3/2*x*y,x)
x*y
TESTS::
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain
sage: R.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex')
sage: P.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex')
sage: P.monomial_lcm(x*y,R.gen())
x*y
::
sage: P.monomial_lcm(P(3/2),P(2/3))
1
::
sage: P.monomial_lcm(x,P(1))
x
"""
one = self.base_ring()(1)
f=f.dict().keys()[0]
g=g.dict().keys()[0]
length = len(f)
res = {}
for i in f.common_nonzero_positions(g):
res[i] = max([f[i],g[i]])
res = self(PolyDict({ETuple(res,length):one},\
force_int_exponents=False,force_etuples=False))
return res
def monomial_all_divisors(self, t):
r"""
Return a list of all monomials that divide ``t``, coefficients are
ignored.
INPUT:
- ``t`` - a monomial.
OUTPUT: a list of monomials.
EXAMPLES::
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain
sage: P.<x,y,z> = MPolynomialRing_polydict_domain(QQ,3, order='degrevlex')
sage: P.monomial_all_divisors(x^2*z^3)
[x, x^2, z, x*z, x^2*z, z^2, x*z^2, x^2*z^2, z^3, x*z^3, x^2*z^3]
ALGORITHM: addwithcarry idea by Toon Segers
"""
def addwithcarry(tempvector, maxvector, pos):
if tempvector[pos] < maxvector[pos]:
tempvector[pos] += 1
else:
tempvector[pos] = 0
tempvector = addwithcarry(tempvector, maxvector, pos + 1)
return tempvector
if not t.is_monomial():
raise TypeError("only monomials are supported")
R = self
one = self.base_ring().one()
M = list()
v, = t.dict()
maxvector = list(v)
tempvector = [
0,
] * len(maxvector)
pos = 0
while tempvector != maxvector:
tempvector = addwithcarry(list(tempvector), maxvector, pos)
M.append(
R(
PolyDict({ETuple(tempvector): one},
force_int_exponents=False,
force_etuples=False)))
return M
def monomial_quotient(self, f, g, coeff=False):
r"""
Return ``f/g``, where both ``f`` and ``g`` are treated as monomials.
Coefficients are ignored by default.
INPUT:
- ``f`` - monomial.
- ``g`` - monomial.
- ``coeff`` - divide coefficients as well (default:
False).
OUTPUT: monomial.
EXAMPLE::
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain
sage: P.<x,y,z> = MPolynomialRing_polydict_domain(QQ, 3, order='degrevlex')
sage: P.monomial_quotient(3/2*x*y, x)
y
::
sage: P.monomial_quotient(3/2*x*y, 2*x, coeff=True)
3/4*y
TESTS::
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain
sage: R.<x,y,z> = MPolynomialRing_polydict_domain(QQ,3, order='degrevlex')
sage: P.<x,y,z> = MPolynomialRing_polydict_domain(QQ,3, order='degrevlex')
sage: P.monomial_quotient(x*y, x)
y
::
sage: P.monomial_quotient(x*y, R.gen())
y
::
sage: P.monomial_quotient(P(0), P(1))
0
::
sage: P.monomial_quotient(P(1), P(0))
Traceback (most recent call last):
...
ZeroDivisionError
::
sage: P.monomial_quotient(P(3/2), P(2/3), coeff=True)
9/4
::
sage: P.monomial_quotient(x, y) # Note the wrong result
x*y^-1
::
sage: P.monomial_quotient(x, P(1))
x
.. note::
Assumes that the head term of f is a multiple of the head
term of g and return the multiplicant m. If this rule is
violated, funny things may happen.
"""
from sage.rings.polynomial.multi_polynomial_element import MPolynomial_polydict
if not f:
return f
if not g:
raise ZeroDivisionError
if not coeff:
coeff = self.base_ring()(1)
else:
coeff = self.base_ring()(f.dict().values()[0] /
g.dict().values()[0])
f = f.dict().keys()[0]
g = g.dict().keys()[0]
res = f.esub(g)
return MPolynomial_polydict(self, PolyDict({res:coeff},\
force_int_exponents=False, \
force_etuples=False))