def radical_singular(I):
r""" Return the radical of this ideal.
INPUT:
- ``I`` -- an ideal of a polynomial ring over `\mathbb{Z}`
OUTPUT: the radical of `I`
"""
singular_lib('primdecint.lib')
return singular_function('radicalZ')(I)
def minimal_ass_primes_singular(I):
r""" Return the list of minimal associated primes of an ideal.
INPUT:
- ``I`` -- an ideal of a polynomial ring over `\mathbb{Z}`
OUTPUT: the list of all minimal associated prime ideals of `I`
"""
singular_lib('primdecint.lib')
R = I.ring()
return [R.ideal(G) for G in singular_function('minAssZ')(I)]
def horizontal_minimal_ass_primes(I):
r""" Return the list of horizontal minimal associated primes of I.
INPUT:
- ``I`` -- an ideal of a polynomial ring over `\mathbb{Z}`
OUTPUT: the list of all horizontal minimal associated prime ideals of `I`.
Here a prime ideal `P` of a polynomial ring `R = \mathbb{Z}[x_1,\ldots,x_n]`
is called *horizontal* if `P\cap\mathbb{Z}=(0)`.
"""
singular_lib('primdecint.lib')
R = I.ring()
P_list = [R.ideal(G) for G in singular_function('minAssZ')(I)]
return [P for P in P_list if P.elimination_ideal(R.gens()).is_zero()]
M.set_row_to_multiple_of_row(u, u, a)
M.add_multiple_of_row(u, i, -b)
#print "\nafter: \n", M
Mt = M[:, -MI.nrows():]
Mres = M[:, :M0.ncols()]
assert (Mt * (apply_swap(M0, swapped)) == Mres)
return Mres, Mt, tuple(swapped)
from sage.libs.singular.function import singular_function
from sage.libs.singular.function import lib as singular_lib
singular_lib('primdecint.lib')
primdecZ = singular_function('primdecZ')
radicalZ = singular_function('radicalZ')
def primdec_ZZ(I):
R = I.ring()
P = primdecZ(I)
V = [R.ideal(X[0]) for X in P]
return V
def radical_ZZ(I):
R = I.ring()
r = radicalZ(I)
return R.ideal(r)
#print "g:", g, "a:", a, "b:", b
M.set_row_to_multiple_of_row(u, u, a)
M.add_multiple_of_row(u,i,-b)
#print "\nafter: \n", M
Mt = M[:,- MI.nrows():]
Mres = M[:,:M0.ncols()]
assert(Mt*(apply_swap(M0,swapped)) == Mres)
return Mres,Mt,tuple(swapped)
from sage.libs.singular.function import singular_function
from sage.libs.singular.function import lib as singular_lib
singular_lib('primdecint.lib')
primdecZ = singular_function('primdecZ')
radicalZ = singular_function('radicalZ')
def primdec_ZZ(I):
R = I.ring()
P = primdecZ(I)
V = [ R.ideal(X[0]) for X in P]
return V
def radical_ZZ(I):
R = I.ring()
r = radicalZ(I)
return R.ideal(r)
class FoundAttack(Exception):