def get_matrix_P1xP1( pol_lst, vars = 'y0,y1,y2,y3' ):
'''
Obtains the matrix M so that wrt monomial basis vector v defined
by SERing.get_mon_P1xP1() we recover pol_lst as M*v.
Parameters
----------
pol_lst : list<SERing.R>
List of homogeneous polynomials of equal bi-degree in
(y0:y1;y2:y3) where the variables are specified by vars.
vars: string
Names of the variables.
Returns
-------
sage_Matrix<SERing.R>
A matrix with polynomials in QQ[a,b,c,d,e,f,g,h].
'''
pol_lst = ring( pol_lst )
m, n = SERing.get_bidegree( pol_lst, vars )
mon_lst = SERing.get_mon_P1xP1( m, n, vars )
SETools.p( 'm = ' + str( m ) + ', n = ' + str( n ) )
mat = []
for pol in pol_lst:
row = []
for mon in mon_lst:
row += [ pol.coefficient( mon ) ]
mat += [row]
return sage_matrix( mat )
def get_matrix_P2( pol_lst, vars = 'x0,x1,x2' ):
'''
Obtains the matrix M so that wrt monomial basis vector v defined
by SERing.get_mon_P1xP1() we recover pol_lst as M*v.
Parameters
----------
pol_lst : list<SERing.R>
List of homogeneous polynomials of equal degree in (x0:x1:x2),
where the variables are specified by vars.
vars: string
Names of the variables.
Returns
-------
sage_Matrix<SERing.R>
The coefficient matrix of pol_lst whose entries are polynomials.
'''
pol_lst = ring( pol_lst )
d = SERing.get_degree( pol_lst, vars )
mon_lst = SERing.get_mon_P2( d, vars )
SETools.p( 'd = ' + str( d ) )
mat = []
for pol in pol_lst:
row = []
for mon in mon_lst:
row += [ pol.coefficient( mon ) ]
mat += [row]
return sage_matrix( mat )
def test__get_matrix_P2(self):
mat = '[(8, -1, 0, 1/17, 2, 0, 1, 4/15, 2, 0), (1, 1/6, -1, 1, -5/7, -3/2, -5, -78, -1, 0), (-1, 0, -3/2, -9/2, -1, -1, 1, -1, 0, -35), (-11/4, 0, 0, 0, -1, 1, 0, 1/2, -5/4, -1/2), (1/7, -7/11, -2/13, 1, -3, 1/2, -1, 1/2, 0, 1/2), (4, -1, -1, 4, 0, 7/4, 3/5, 0, 1, -35), (-1/5, 0, -2, -55/2, -1/3, 2, -2, -2/11, -1, -1/2), (-1/6, -1/2, 2, 0, 2, 2, 0, -1, -1, 8), (-1, -2/3, 5/33, 2, 0, -1/48, -1, -1/15, 1/2, 1), (0, 3/2, 0, -2, 1/5, -4/7, 1/2, 1/4, 1, 0)]'
mat = sage_matrix(sage_QQ, ring(mat))
mon_lst = SERing.get_mon_P2(3)
pol_lst = list(mat * sage_vector(mon_lst))
out = SERing.get_matrix_P2(pol_lst)
print(mat)
print(mon_lst)
print(pol_lst)
print(out)
assert mat == out
def test__get_matrix_P1P1(self):
mat = '[(13, 1, -2, 1/2, 1, 3, 3, 0, -1), (2/5, -2/5, -2, -1, -2, -1, -5/49, 19, 0), (1, -1, 0, -3/2, 1/2, -1/22, 2, 5, -8/9), (7/23, -1/2, 0, -1, 1, 32, 1/2, 0, 1)]'
mat = sage_matrix(sage_QQ, ring(mat))
mon_lst = SERing.get_mon_P1xP1(2, 2)
pol_lst = list(mat * sage_vector(mon_lst))
out = SERing.get_matrix_P1xP1(pol_lst)
print(mat)
print(mon_lst)
print(pol_lst)
print(out)
assert mat == out
def random_matrix_QQ( m, n ):
'''
Parameters
----------
m : int
n : int
Returns
-------
sage_matrix<sage_QQ>
A random mxn-matrix with rational number as entries.
'''
mat = []
for ri in range( m ):
row = []
for ci in range( n ):
row += [ SERing.random_QQ() ]
mat += [row]
mat = sage_matrix( sage_QQ, mat )
return mat
def random_inv_matrix( m ):
'''
Parameters
----------
m : int
Returns
-------
sage_matrix<sage_QQ>
A random invertible m x m-matrix with either -1, 0 or 1 as entries.
'''
while True:
mat = []
for ri in range( m ):
row = []
for ci in range( m ):
row += [ SERing.random_ZZ( 1 ) ]
mat += [row]
mat = sage_matrix( sage_QQ, mat )
if mat.is_invertible():
return mat
def usecase_B5():
'''
We compute the projective isomorphism between two
conic bundles that are parametrized by the
birational maps
ff: P2 ---> X and gg: P1xP1 ---> Y
Further explanation of this example can be found
in the accompanying arxiv article on projective
isomorphisms between rational surfaces.
'''
# we construct linear series associated to ff in order to determine
# the generators of the graded coordinate ring of conic bundle X
# basepoints in chart x0!=0;
p1 = (0, 0)
p2 = (0, 1)
p3 = (1, 0)
# 0f+p = e0-e1
PolyRing.reset_base_field()
bp_tree = BasePointTree()
bp_tree.add('z', p1, 1)
f0p1 = SERing.conv(LinearSeries.get([1], bp_tree).pol_lst)
SETools.p('f0p1 =', len(f0p1), f0p1)
# 1f-3p = e0+e1-e2-e3
bp_tree = BasePointTree()
bp_tree.add('z', p2, 1)
bp_tree.add('z', p3, 1)
f1m3 = SERing.conv(LinearSeries.get([1], bp_tree).pol_lst)
SETools.p('f1m3 =', len(f1m3), f1m3)
# 1f-2p = 2e0-e2-e3
bp_tree = BasePointTree()
bp_tree.add('z', p2, 1)
bp_tree.add('z', p3, 1)
f1m2 = SERing.conv(LinearSeries.get([2], bp_tree).pol_lst)
SETools.p('f1m2 =', len(f1m2), f1m2)
# 1f-1p = 3e0-e1-e2-e3
bp_tree = BasePointTree()
bp_tree.add('z', p1, 1)
bp_tree.add('z', p2, 1)
bp_tree.add('z', p3, 1)
f1m1 = SERing.conv(LinearSeries.get([3], bp_tree).pol_lst)
SETools.p('f1m1 =', len(f1m1), f1m1)
# 1f-0p = 4e0-2e1-e2-e3
bp_tree = BasePointTree()
bp_tree.add('z', p1, 2)
bp_tree.add('z', p2, 1)
bp_tree.add('z', p3, 1)
f1m0 = SERing.conv(LinearSeries.get([4], bp_tree).pol_lst)
SETools.p('f1m0 =', len(f1m0), f1m0)
# 2f-4p = 4e0-2e2-2e3
bp_tree = BasePointTree()
bp_tree.add('z', p2, 2)
bp_tree.add('z', p3, 2)
f2m4 = SERing.conv(LinearSeries.get([4], bp_tree).pol_lst)
SETools.p('f2m4 =', len(f2m4), f2m4)
# by inspection we recover the generators of graded ring of ff
U = ring('x1'), ring('x2'), ring('x1+x2-x0'), ring('x1*x2'), ring(
'(x1+x2-x0)^2')
# compute bidegree (2,d) in order to find a relation between the generators
u = u0, u1, u2, u3, u4 = ring('u0,u1,u2,u3,u4')
SETools.p(
'Compare number of monomials of given bi-weight with dimension predicted by the Riemann-Roch formula...'
)
for d in reversed([-i for i in range(8)]):
w_lst = [(0, 1), (0, 1), (1, -3), (1, -2), (1, -2)]
SETools.p('\tweight=', (2, d), ',\t#monomials=',
len(SERing.get_wmon_lst(u, w_lst, 2, d)), ',\tRR=',
29 + 5 * d)
# template for generators of coordinate ring for weight (2,-1) and (1,0)
T2m4 = ring(
'[u3^2,u3*u4,u4^2,u0*u2*u3,u0*u2*u4,u1*u2*u3,u1*u2*u4,u0^2*u2^2,u0*u1*u2^2,u1^2*u2^2]'
)
T1m0 = ring(
'[u1^2*u4,u1^2*u3,u1^3*u2,u0*u1*u4,u0*u1*u3,u0*u1^2*u2,u0^2*u4,u0^2*u3,u0^2*u1*u2,u0^3*u2]'
)
SETools.p('T2m4 =', T2m4)
SETools.p('T1m0 =', T1m0)
# find linear relation for f2m4
a = a0, a1, a2, a3, a4, a5, a6, a7, a8, a9 = [
elt.subs({u[i]: U[i]
for i in range(5)}) for elt in T2m4
] # @UnusedVariable
mata = sage_matrix(sage_QQ, SERing.get_matrix_P2(a))
kera = mata.transpose().right_kernel().matrix()
SETools.p('kera =', kera)
assert kera * sage_vector(a) == sage_vector([0])
assert a1 - a8 == 0
# construct map gg from ff
# sage_Permutations(10).random_element().to_matrix().rows()
ff = f1m0
matp = [(0, 0, 0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 1, 0, 0),
(0, 0, 0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 1),
(0, 0, 0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0, 0, 0),
(0, 0, 0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 1, 0, 0, 0, 0, 0, 0, 0),
(0, 1, 0, 0, 0, 0, 0, 0, 0, 0), (1, 0, 0, 0, 0, 0, 0, 0, 0, 0)]
matp = sage_matrix(matp)
x0, x1, x2, y0, y1, y2, y3 = ring(
'x0,x1,x2,y0,y1,y2,y3') # P2(x0:x1:x2) and P1xP1(y0:y1;y2:y3)
gg = [comp.subs({x0: y1 * y3, x1: y0 * y2, x2: y1 * y2}) for comp in ff]
gg = list(matp * sage_vector(gg))
gcd_gg = sage_gcd(gg)
gg = [comp / gcd_gg for comp in gg]
SETools.p('gcd_gg =', gcd_gg)
SETools.p('ff =', len(ff), ff)
SETools.p('gg =', len(gg), gg)
# we construct linear series associated to gg in order to determine
# the generators of the graded coordinate ring of conic bundle Y
# determine and set basepoint tree
ls = LinearSeries(SERing.conv(gg), PolyRing('x,y,v,w'))
bp_tree = ls.get_bp_tree()
SETools.p('bp_tree(gg) =', bp_tree)
tree_211 = BasePointTree(['xv', 'xw', 'yv', 'yw'])
tree_211.add('xw', (0, 0), 2).add('t', (1, 0), 1)
tree_211.add('yv', (0, 1), 1)
# 1g+0q = 4l0+2l1-2e1-e2-e3
g1m0 = SERing.conv(LinearSeries.get([4, 2], tree_211).pol_lst)
SETools.p('g1m0 =', len(g1m0), g1m0)
# 1g-3q = (l0+l1-e1-e2-e3) + (b-e1)
g1m3 = SERing.conv(LinearSeries.get([1, 2], tree_211).pol_lst)
SETools.p('g1m3 =', len(g1m3), g1m3)
# 1g-2q = 2l0+2l1-2e1-e2-e3
g1m2 = SERing.conv(LinearSeries.get([2, 2], tree_211).pol_lst)
SETools.p('g1m2 =', len(g1m2), g1m2)
# 1g-1q = 3l0+2l1-2e1-e2-e3
g1m1 = SERing.conv(LinearSeries.get([3, 2], tree_211).pol_lst)
SETools.p('g1m1 =', len(g1m1), g1m1)
# 2g-4q = 4l0+4l1-4e1-2e2-2e3
tree_422 = BasePointTree(['xv', 'xw', 'yv', 'yw'])
tree_422.add('xw', (0, 0), 4).add('t', (1, 0), 2)
tree_422.add('yv', (0, 1), 2)
g2m4 = SERing.conv(LinearSeries.get([4, 4], tree_422).pol_lst)
SETools.p('g2m4 =', len(g2m4), g2m4)
# by inspection we recover the generators of graded ring of gg
V = ring('y0'), ring('y1'), ring('y0*y2^2+y1*y2^2-y1*y2*y3'), ring(
'y0*y1*y2^2'), ring('y0^2*y2^2+y1^2*y2^2-y1^2*y3^2')
# find linear relation for g2m4
b = b0, b1, b2, b3, b4, b5, b6, b7, b8, b9 = [
elt.subs({u[i]: V[i]
for i in range(5)}) for elt in T2m4
] # @UnusedVariable
matb = sage_matrix(sage_QQ, SERing.get_matrix_P1xP1(b))
kerb = matb.transpose().right_kernel().matrix()
SETools.p('kerb =', kerb)
assert kerb * sage_vector(b) == sage_vector([0])
assert 2 * b0 + b1 - 2 * b3 - 2 * b5 + b8 == 0
# compute inverse of G
G = [elt.subs({u[i]: V[i] for i in range(5)}) for elt in T1m0]
z = ring('z0,z1,z2,z3,z4,z5,z6,z7,z8,z9')
t = ring('t')
ide = [G[i] * z[0] - z[i] * G[0] for i in range(10)] + [t * G[0] - 1]
I01 = sage_ideal(ide).elimination_ideal([t, y2, y3]).gens()
I23 = sage_ideal(ide).elimination_ideal([t, y0, y1]).gens()
I01 = [elt for elt in I01
if elt.degree(y0) == 1 and elt.degree(y1) == 1][0]
I23 = [elt for elt in I23
if elt.degree(y2) == 1 and elt.degree(y3) == 1][0]
Q0 = I01.coefficient(y1)
Q1 = -I01.coefficient(y0)
Q2 = I23.coefficient(y3)
Q3 = -I23.coefficient(y2)
Q = [Q0, Q1, Q2, Q3]
SETools.p('Q =', Q) # [-z9, -z8, -z8, -z6 - 2*z7 + z8 + z9]
# check the inverse
QoG = [q.subs({z[i]: G[i] for i in range(10)}) for q in Q]
gcd01 = sage_gcd(QoG[0], QoG[1])
gcd23 = sage_gcd(QoG[2], QoG[3])
QoG = [QoG[0] / gcd01, QoG[1] / gcd01, QoG[2] / gcd23, QoG[3] / gcd23]
SETools.p('QoG =', QoG)
assert QoG == [y0, y1, y2, y3]
# compose F with projective isomorphism P
c = c0, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12 = [
ring('c' + str(i)) for i in range(13)
]
dctZ = {u[i]: z[i] for i in range(5)}
dctP = {
z[0]: c0 * u0 + c1 * u1,
z[1]: c2 * u0 + c3 * u1,
z[2]: c4 * u2,
z[3]: c5 * u3 + c6 * u4 + c7 * u0 * u2 + c8 * u1 * u2,
z[4]: c9 * u3 + c10 * u4 + c11 * u0 * u2 + c12 * u1 * u2
}
PoF = [comp.subs(dctZ).subs(dctP) for comp in T1m0]
PoF = [comp.subs({u[i]: U[i] for i in range(5)}) for comp in PoF]
PoF = [comp / sage_gcd(PoF) for comp in PoF]
SETools.p('PoF =', len(PoF), PoF)
# compose PoF with Q
QoPoF = [comp.subs({z[i]: PoF[i] for i in range(10)}) for comp in Q]
gcd01 = sage_gcd(QoPoF[0], QoPoF[1])
gcd23 = sage_gcd(QoPoF[2], QoPoF[3])
QoPoF = [
QoPoF[0] / gcd01, QoPoF[1] / gcd01, QoPoF[2] / gcd23, QoPoF[3] / gcd23
]
SETools.p('QoPoF =', len(QoPoF), QoPoF)
# create a list of equations for the ci
b = T2m4
rel_g4m2 = 2 * b[0] + b[1] - 2 * b[3] - 2 * b[5] + b[8]
SETools.p('rel_g4m2 =', rel_g4m2)
rel_g4m2 = rel_g4m2.subs(dctZ).subs(dctP).subs(
{u[i]: U[i]
for i in range(5)})
SETools.p('rel_g4m2 =', rel_g4m2)
rel_lst = []
x = ring('[x0,x1,x2]')
for exp in sage_Compositions(4 + 3, length=3):
rel_lst += [rel_g4m2.coefficient({x[i]: exp[i] - 1 for i in range(3)})]
SETools.p('rel_lst =', len(rel_lst), rel_lst)
t = ring('t')
rel_lst += [(c0 * c3 - c1 * c2) * c4 * (c5 * c10 - c9 * c6) * t - 1]
# solve for ci and put the solutions in dictionary form
prime_lst = sage_ideal(rel_lst).elimination_ideal(
t).primary_decomposition()
SETools.p('prime_lst =', len(prime_lst))
for gen_lst in [prime.gens() for prime in prime_lst]:
sol_dct = sage_solve([sage_SR(gen) for gen in gen_lst],
[sage_SR(elt) for elt in c],
solution_dict=True)
assert len(sol_dct) == 1
SETools.p('\t gen_lst =', gen_lst)
SETools.p('\t sol_dct =', sol_dct[0])
prime_lst2 = []
prime_lst2 += [prime_lst[0].gens() + [c0 - 1, c4 - 1]]
prime_lst2 += [prime_lst[1].gens() + [c1 - 1, c4 - 1]]
prime_lst2 += [prime_lst[2].gens() + [c1 - 1, c4 - 1]]
prime_lst2 += [prime_lst[3].gens() + [c0 - 1, c4 - 1]]
SETools.p('Added equations to prime_lst to simplify solutions:')
for gen_lst in prime_lst2:
sol_dct = sage_solve([sage_SR(gen) for gen in gen_lst],
[sage_SR(elt) for elt in c],
solution_dict=True)
assert len(sol_dct) == 1
SETools.p('\t gen_lst =', gen_lst)
SETools.p('\t sol_dct =', sol_dct[0])
r0, r1 = ring('r0,r1')
sol0 = {
c0: 1,
c1: 0,
c2: 0,
c3: -r0 * r1,
c4: 1,
c5: 0,
c6: r0,
c7: 0,
c8: 0,
c9: r1,
c10: -2 * r0,
c11: 2,
c12: -2 * r0 * r1
}
sol1 = {
c0: 0,
c1: 1,
c2: -r0 * r1,
c3: 0,
c4: 1,
c5: 0,
c6: r0,
c7: 0,
c8: 0,
c9: r1,
c10: -2 * r0,
c11: -2 * r0 * r1,
c12: 2
}
sol2 = {
c0: 0,
c1: 1,
c2: -r0 * r1,
c3: 0,
c4: 1,
c5: r0,
c6: 0,
c7: 0,
c8: 0,
c9: -2 * r0,
c10: r1,
c11: -2 * r0 * r1,
c12: 2
}
sol3 = {
c0: 1,
c1: 0,
c2: 0,
c3: -r0 * r1,
c4: 1,
c5: r0,
c6: 0,
c7: 0,
c8: 0,
c9: -2 * r0,
c10: r1,
c11: 2,
c12: -2 * r0 * r1
}
sol_lst = [sol0, sol1, sol2, sol3]
SETools.p('Simplified solutions by hand:')
for sol in sol_lst:
SETools.p('\t', sol)
# compose compatible reparametrizations with gg
y = ring('[y0,y1,y2,y3]')
gr_lst = []
SETools.p('Computing (gg o r) for each sol in sol_lst...')
for sol in sol_lst:
gr = [
comp.subs({y[i]: QoPoF[i]
for i in range(4)}).subs(sol) for comp in gg
]
SETools.p('\t gr =', gr)
gcd_gr = sage_gcd(gr)
SETools.p('\t\t gcd_gr =', gcd_gr)
gr_lst += [[comp / gcd_gr for comp in gr]]
SETools.p('\t\t gr/gcd_gr =', gr_lst[-1])
SETools.p('gr_lst =', len(gr_lst))
for gr in gr_lst:
SETools.p('\t gr =', gr)
# get coefficient matrix of ff and its kernel
mff = SERing.get_matrix_P2(ff)
kff = mff.right_kernel_matrix().T
SETools.p('mff =', mff.dimensions(), list(mff))
SETools.p('kff =', kff.dimensions(), list(kff))
assert (mff * kff).is_zero()
# get implicit equations for image of gg
z = ring('z0,z1,z2,z3,z4,z5,z6,z7,z8,z9')
y = ring('y0,y1,y2,y3')
igg = SERing.R.ideal([z[i] - gg[i] for i in range(10)
]).elimination_ideal([y[i] for i in range(4)])
SETools.p('igg =', list(igg.gens()))
# Compute isomorphisms for each gr
SETools.p('Compute projective isomorphism for each gr in gr_lst:')
for gr in gr_lst:
mgr = SERing.get_matrix_P2(gr)
mgk = mgr * kff
assert mgk.is_zero() # because the surfaces in P^9 are linearly normal
Ef = sage_matrix(sage_QQ, mff.rows() + kff.T.rows())
Egr = sage_matrix(mgr.rows() + kff.T.rows())
UpI = Egr * Ef.inverse()
assert (UpI.submatrix(10, 10) - sage_identity_matrix(5)).is_zero()
U = UpI.submatrix(0, 0, 10, 10)
SETools.p('\tU =', U.dimensions(), list(U))
# check if the answer is correct by substituting into the equations of Y
Uff = list(U * sage_vector(ff))
iggs = igg.subs({z[i]: Uff[i] for i in range(10)})
assert iggs.is_zero()
def usecase_B4():
'''
We compute the projective automorphism of the
rational normal scrolls that is parametrized
the birational map f: P2 ---> X.
Further explanation of this example can be found
in the accompanying arxiv article on projective
isomorphisms between rational surfaces.
'''
# e0-e1
p1 = (0, 0)
p2 = (1, 0)
p3 = (0, 1)
PolyRing.reset_base_field()
bpt = BasePointTree()
bpt.add('z', p1, 1)
f0p1 = SERing.conv(LinearSeries.get([1], bpt).pol_lst)
SETools.p('f0p1 =', len(f0p1), f0p1)
# e0-e2-e3
bpt = BasePointTree()
bpt.add('z', p2, 1)
bpt.add('z', p3, 1)
f1m2 = SERing.conv(LinearSeries.get([1], bpt).pol_lst)
SETools.p('f1m2 =', len(f1m2), f1m2)
# 2e0-e1-e2-e3
bpt = BasePointTree()
bpt.add('z', p1, 1)
bpt.add('z', p2, 1)
bpt.add('z', p3, 1)
f1m1 = SERing.conv(LinearSeries.get([2], bpt).pol_lst)
SETools.p('f1m1 =', len(f1m1), f1m1)
# 3e0-2e1-e2-e3
bpt = BasePointTree()
bpt.add('z', p1, 2)
bpt.add('z', p2, 1)
bpt.add('z', p3, 1)
f1m0 = SERing.conv(LinearSeries.get([3], bpt).pol_lst)
SETools.p('f1m0 =', len(f1m0), f1m0)
# set generators for the graded coordinate ring of f
U = [ring('x1'), ring('x2'), ring('x1+x2-x0'), ring('x1*x2')]
u = ring('u0,u1,u2,u3')
# obtain monomials of weight (1,0)
w_lst = [(0, 1), (0, 1), (1, -2), (1, -1)]
M1m0 = SERing.get_wmon_lst(u, w_lst, 1, 0)
SETools.p('M1m0 =', len(M1m0), M1m0)
# compose F with projective isomorphism P
z = [ring('z' + str(i)) for i in range(6)]
c = [ring('c' + str(i)) for i in range(8)]
dctZ = {u[i]: z[i] for i in range(4)}
dctP = {
z[0]: c[0] * u[0] + c[1] * u[1],
z[1]: c[2] * u[0] + c[3] * u[1],
z[2]: c[4] * u[2],
z[3]: c[5] * u[3] + c[6] * u[0] * u[2] + c[7] * u[1] * u[2]
}
PoF = [comp.subs(dctZ).subs(dctP) for comp in M1m0]
SETools.p('PoF wrt u =', len(PoF), PoF)
PoF = [comp.subs({u[i]: U[i] for i in range(4)}) for comp in PoF]
PoF = [comp / sage_gcd(PoF) for comp in PoF]
SETools.p('PoF =', len(PoF), PoF)
# recover matrix for automorphism P
F = [comp.subs({u[i]: U[i] for i in range(4)}) for comp in M1m0]
M = []
for pol in [comp.subs(dctZ).subs(dctP) for comp in M1m0]:
row = []
for mon in M1m0:
row += [pol.coefficient(mon)]
M += [row]
M = sage_matrix(M)
SETools.p('M =', M.dimensions(), '\n' + str(M))
MF = list(M * sage_vector(F))
assert MF == PoF
# compute the inverse Q of F
t = ring('t')
x = ring('x0,x1,x2')
ide = [F[i] * z[0] - z[i] * F[0] for i in range(5)] + [t * F[0] - 1]
I1 = sage_ideal(ide).elimination_ideal([t, x[2]]).gens()
I2 = sage_ideal(ide).elimination_ideal([t, x[1]]).gens()
I1 = [
elt for elt in I1 if elt.degree(x[0]) == 1 and elt.degree(x[1]) == 1
][0]
I2 = [
elt for elt in I2 if elt.degree(x[0]) == 1 and elt.degree(x[2]) == 1
][0]
Q0 = I1.coefficient(x[1])
Q1 = -I1.coefficient(x[0])
Q2 = I2.coefficient(x[0])
Q = [Q0, Q1, Q2]
SETools.p('Q =', Q)
QoF = [comp.subs({z[i]: F[i] for i in range(5)}) for comp in Q]
QoF = [comp / sage_gcd(QoF) for comp in QoF]
assert QoF == [x[0], x[1], x[2]]
# compute the composition
QoPoF = [comp.subs({z[i]: PoF[i] for i in range(5)}) for comp in Q]
QoPoF = [comp / sage_gcd(QoPoF) for comp in QoPoF]
SETools.p('QoPoF =', len(QoPoF), QoPoF)
# from the compatible reparametrizations QoPoF we compute the
# projective automorphisms U of X.
f = f1m0
gr = [comp.subs({x[i]: QoPoF[i] for i in range(3)}) for comp in f]
gcd_gr = sage_gcd(gr)
gr = [comp / gcd_gr for comp in gr]
Mf = SERing.get_matrix_P2(f)
Mgr = SERing.get_matrix_P2(gr)
Kf = Mf.right_kernel_matrix().T
SETools.p('f =', len(f), f)
SETools.p('gr =', len(gr), gr)
SETools.p('\t gcd_gr =', gcd_gr)
SETools.p('Mf =', Mf.dimensions(), list(Mf))
SETools.p('Mgr =', Mgr.dimensions(), list(Mgr))
SETools.p('Kf =', Kf.dimensions(), list(Kf))
assert (Mf * Kf).is_zero()
assert (Mgr * Kf).is_zero()
Ef = sage_matrix(sage_QQ, list(Mf) + list(Kf.T))
Egr = sage_matrix(list(Mgr) + list(Kf.T))
UpI = Egr * ~Ef
assert (UpI.submatrix(5, 5) - sage_identity_matrix(5)).is_zero()
U = UpI.submatrix(0, 0, 5, 5)
SETools.p('UpI =', UpI.dimensions(), list(UpI))
SETools.p('U =', U.dimensions(), list(U), '\n' + str(U))
# verify whether U*f is a parametrization for X for all (c0,...,c7)
Uf = list(U * sage_vector(f))
SETools.p('Uf =', len(Uf), Uf)
eqX = sage_ideal([z[i] - f[i]
for i in range(5)]).elimination_ideal([x[0], x[1],
x[2]]).gens()
eqXs = [eq.subs({z[i]: Uf[i] for i in range(5)}) for eq in eqX]
SETools.p('eqX =', len(eqX), eqX)
SETools.p('eqXs=', len(eqXs), eqXs)
assert eqXs == [0, 0, 0]
def usecase_B2():
'''
We compute the projective isomorphisms between
the images of birational maps:
f:P2--->X and g:P1xP1--->Y
Further explanation of this example can be found
in the accompanying arxiv article on projective
isomorphisms between rational surfaces.
'''
x = [ring('x' + str(i)) for i in range(3)]
y = [ring('y' + str(i)) for i in range(4)]
z = [ring('z' + str(i)) for i in range(4)]
c = [ring('c' + str(i)) for i in range(8)]
# the maps f and g are parametrizations of (linear projections of) toric surfaces
f = ring(
'[x0^6*x1^2,x0*x1^5*x2^2,x1^3*x2^5,x0^5*x2^3+x0^5*x2^3+x0^5*x1*x2^2]')
g = ring(
'[y0^3*y1^2*y2^5,y1^5*y2^3*y3^2,y0^2*y1^3*y3^5,y0^5*y2^2*y3^3+y0^4*y1*y2^3*y3^2]'
)
g = [g[0], g[1] + g[0], g[2], g[3] + g[2]]
SETools.p('f =', len(f), f)
SETools.p('g =', len(g), g)
assert sage_gcd(f) == 1
assert sage_gcd(g) == 1
# we compute the implicit equations of the images of the maps f and g
eqf = sage_ideal([z[i] - f[i]
for i in range(4)]).elimination_ideal(x).gens()
SETools.p('eqf =', eqf)
assert len(eqf) == 1
assert eqf[0].degree() == 26
eqg = sage_ideal([z[i] - g[i]
for i in range(4)]).elimination_ideal(y).gens()
SETools.p('eqg =', eqg)
assert len(eqg) == 1
assert eqg[0].degree() == 26
# We compute the coefficient matrix Mf and its kernel Kf
Mf = SERing.get_matrix_P2(f)
Kf = Mf.right_kernel_matrix().T
SETools.p('Mf =', Mf.dimensions(), list(Mf))
SETools.p('Kf =', Kf.dimensions(), list(Kf))
assert (Mf * Kf).is_zero()
# we do a basepoint analysis for f and g
bf = LinearSeries(SERing.conv(f), PolyRing('x,y,z', True)).get_bp_tree()
SETools.p('bf =', bf)
bg = LinearSeries(SERing.conv(g), PolyRing('x,y,v,w', True)).get_bp_tree()
SETools.p('bg =', bg)
###################################################
# Computing compatible reductions of f and g #
###################################################
# r0-reduction of f and g
hf = LinearSeries.get([8], bf)
hg = LinearSeries.get([5, 5], bg)
SETools.p(len(hf.pol_lst), SERing.conv(hf.pol_lst))
SETools.p(len(hg.pol_lst), SERing.conv(hg.pol_lst))
assert len(hf.pol_lst) == len(hg.pol_lst) == 16
# r1-reduction of hf and hg
hft = BasePointTree()
hft.add('x', (0, 0), 2).add('t', (0, 0), 1)
hft.add('y', (0, 0), 2).add('t', (0, 0), 1)
hft.add('z', (0, 0), 1)
hf = LinearSeries.get([5], hft)
hgt = BasePointTree(['xv', 'xw', 'yv', 'yw'])
hgt.add('xv', (0, 0), 1)
hgt.add('xw', (0, 0), 1)
hgt.add('yv', (0, 0), 1)
hgt.add('yw', (0, 0), 1)
hg = LinearSeries.get([3, 3], hgt)
SETools.p(len(hf.pol_lst), SERing.conv(hf.pol_lst))
SETools.p(len(hg.pol_lst), SERing.conv(hg.pol_lst))
assert len(hf.pol_lst) == len(hg.pol_lst) == 12
# second r1-reduction of hf and hg
hft = BasePointTree()
hft.add('x', (0, 0), 1)
hft.add('y', (0, 0), 1)
hf = LinearSeries.get([2], hft)
hg = LinearSeries(['x', 'y', 'v', 'w'], PolyRing('x,y,v,w'))
SETools.p(len(hf.pol_lst), SERing.conv(hf.pol_lst))
SETools.p(len(hg.pol_lst), SERing.conv(hg.pol_lst))
assert len(hf.pol_lst) == len(hg.pol_lst) == 4
###################################################
# Computing the projective isomorphisms #
###################################################
# we compute maps to P1xP1 from two pencils
PolyRing.reset_base_field()
bpt = BasePointTree()
bpt.add('y', (0, 0), 1)
pen1 = SERing.conv(LinearSeries.get([1], bpt).pol_lst)
SETools.p('pen1 =', pen1)
assert set([x[0], x[1]]) == set(pen1)
# thus the first pencil defines a map pen1: (x0:x1:x2) |--> (x0:x1)
bpt = BasePointTree()
bpt.add('x', (0, 0), 1)
pen2 = SERing.conv(LinearSeries.get([1], bpt).pol_lst)
SETools.p('pen2 =', pen2)
assert set([x[0], x[2]]) == set(pen2)
# thus the second pencil defines a map pen2: (x0:x1:x2) |--> (x0:x2)
# We find that
# pen1 x pen2: P2-->P1xP1, (x0:x1:x2) |--> (x0:x1;x0:x2) and
# pen2 x pen1: P2-->P1xP1, (x0:x1:x2) |--> (x0:x2;x0:x1)
# we find the following compatible reparametrizations
# by composing the maps pen1 x pen2 and pen2 x pen1
# with a parametrized map in the identity component of Aut(P1xP1).
r0 = {
y[0]: c[0] * x[0] + c[1] * x[1],
y[1]: c[2] * x[0] + c[3] * x[1],
y[2]: c[4] * x[0] + c[5] * x[2],
y[3]: c[6] * x[0] + c[7] * x[2]
}
r1 = {
y[0]: c[0] * x[0] + c[1] * x[2],
y[1]: c[2] * x[0] + c[3] * x[2],
y[2]: c[4] * x[0] + c[5] * x[1],
y[3]: c[6] * x[0] + c[7] * x[1]
}
# Remark: all substitutions with .subs(...) are performed at the same time.
###################################################
# reparametrization r0 #
###################################################
# compose g with reparametrization r0
gcd0 = sage_gcd([comp.subs(r0) for comp in g])
assert gcd0 == 1
gr0 = [comp.subs(r0) / gcd0 for comp in g]
SETools.p('gr0 =', len(gr0), gcd0, gr0)
assert SERing.get_degree(gr0) == 10
assert SERing.get_degree(f) == 8
# find conditions on c so that gr0 has the same basepoints as f
eqn0_lst = usecase_B2_helper_bp(gr0)
eqn0_lst += [ring('(c0*c3-c1*c2)*(c4*c7-c5*c6)*t-1')]
prime0_lst = sage_ideal(eqn0_lst).elimination_ideal(
ring('t')).primary_decomposition()
SETools.p('eqn0_lst =', len(eqn0_lst), eqn0_lst)
for prime0 in prime0_lst:
SETools.p('\t', prime0.gens())
sol00 = {
c[1]: 0,
c[2]: 0,
c[5]: 0,
c[6]: 0,
c[0]: 1,
c[4]: 1
} # notice that wlog c0=c4=1
sol01 = {
c[0]: 0,
c[3]: 0,
c[4]: 0,
c[7]: 0,
c[1]: 1,
c[5]: 1
} # notice that wlog c1=c5=1
assert len(prime0_lst) == 2
assert set([gen.subs(sol00) for gen in prime0_lst[0].gens()]) == set([0])
assert set([gen.subs(sol01) for gen in prime0_lst[1].gens()]) == set([0])
# sol00: notice that c3!=0 and c7!=0
gcd00 = sage_gcd([comp.subs(sol00) for comp in gr0])
assert gcd00 == x[0] * x[0]
gr00 = [comp.subs(sol00) / gcd00 for comp in gr0]
SETools.p('gr00 =', len(gr00), gcd00, gr00)
assert SERing.get_degree(gr00) == 8
Mgr00 = SERing.get_matrix_P2(gr00)
assert Mgr00.dimensions() == (4, 45)
# find conditions for c so that Mgr00 has the same kernel as the matrix of f
p00_lst = sage_ideal((Mgr00 * Kf).list() +
[ring('c3*c7*t-1')]).elimination_ideal(
ring('t')).primary_decomposition()
assert [p00.gens() for p00 in p00_lst] == [[2 * c[3] - c[7]]]
Mgr00 = Mgr00.subs({c[7]: 2 * c[3]})
SETools.p('Mgr00 =', Mgr00.dimensions(), list(Mgr00))
# found a solution: Mgr00
# sol01: notice that c2!=0 and c6!=0
gcd01 = sage_gcd([comp.subs(sol01) for comp in gr0])
assert gcd01 == x[0] * x[0]
gr01 = [comp.subs(sol01) / gcd01 for comp in gr0]
SETools.p('gr01 =', len(gr01), gcd01, gr01)
assert SERing.get_degree(gr01) == 8
assert [] == sage_ideal((SERing.get_matrix_P2(gr01) * Kf).list() +
[ring('c2*c6*t-1')]).elimination_ideal(
ring('t')).primary_decomposition()
# --> no solution
###################################################
# reparametrization r1 #
###################################################
# compose g with reparametrization r1
gcd1 = sage_gcd([comp.subs(r1) for comp in g])
assert gcd1 == 1
gr1 = [comp.subs(r1) / gcd1 for comp in g]
SETools.p('gr1 =', gcd1, gr1)
assert SERing.get_degree(gr1) == 10
assert SERing.get_degree(f) == 8
# find conditions on c so that gr1 has the same basepoints as f
eqn1_lst = usecase_B2_helper_bp(gr1)
eqn1_lst += [ring('(c0*c3-c1*c2)*(c4*c7-c5*c6)*t-1')]
SETools.p('eqn1_lst =', len(eqn1_lst), eqn1_lst)
prime1_lst = sage_ideal(eqn1_lst).elimination_ideal(
ring('t')).primary_decomposition()
for prime1 in prime1_lst:
SETools.p('\t', prime1.gens())
sol10 = {
c[0]: 0,
c[3]: 0,
c[5]: 0,
c[6]: 0,
c[1]: 1,
c[4]: 1
} # notice that wlog c1=c4=1
sol11 = {
c[1]: 0,
c[2]: 0,
c[4]: 0,
c[7]: 0,
c[0]: 1,
c[5]: 1
} # notice that wlog c0=c5=1
assert len(prime1_lst) == 2
assert set([gen.subs(sol10) for gen in prime1_lst[0].gens()]) == set([0])
assert set([gen.subs(sol11) for gen in prime1_lst[1].gens()]) == set([0])
# sol10: notice that c2!=0 and c7!=0
gcd10 = sage_gcd([comp.subs(sol10) for comp in gr1])
assert gcd10 == x[0] * x[0]
gr10 = [comp.subs(sol10) / gcd10 for comp in gr1]
SETools.p('gr10 =', len(gr10), gcd10, gr10)
assert SERing.get_degree(gr10) == 8
assert [] == sage_ideal((SERing.get_matrix_P2(gr10) * Kf).list() +
[ring('c2*c7*t-1')]).elimination_ideal(
ring('t')).primary_decomposition()
# --> no solution
# sol11: notice that c3!=0 and c6!=0
gcd11 = sage_gcd([comp.subs(sol11) for comp in gr1])
assert gcd11 == x[0] * x[0]
gr11 = [comp.subs(sol11) / gcd11 for comp in gr1]
SETools.p('gr11 =', len(gr11), gcd11, gr11)
assert SERing.get_degree(gr11) == 8
assert [] == sage_ideal((SERing.get_matrix_P2(gr11) * Kf).list() +
[ring('c3*c6*t-1')]).elimination_ideal(
ring('t')).primary_decomposition()
# --> no solution
###################################################
# compute extended matrices #
###################################################
# Mgr00 is the only case we have to consider as other cases had no solution
Mgr = Mgr00
# compute the projective isomorphism between the images of f and g
# in terms of parametrized matrix U
Ef = sage_matrix(sage_QQ, list(Mf) + list(Kf.T))
Egr = sage_matrix(list(Mgr) + list(Kf.T))
UpI = Egr * ~Ef
assert (UpI.submatrix(4, 4) - sage_identity_matrix(41)).is_zero()
U = UpI.submatrix(0, 0, 4, 4)
U = U / sage_gcd(U.list())
SETools.p('U =', U.dimensions(), list(U), '\n' + str(U))
# verify whether U*f is a parametrization for X for all (c0,...,c7)
Uf = list(U * sage_vector(f))
eqg_sub = [eq.subs({z[i]: Uf[i] for i in range(4)}) for eq in eqg]
if eqg_sub != [0]:
SETools.p('Uf =', len(Uf), Uf)
SETools.p('eqg_sub=', len(eqg_sub), eqg_sub)
assert eqg_sub == [0]
def usecase_B1():
'''
We compute the automorphisms of a Roman surface from a set of
compatible reparametrizations.
If the domain the projective plane, then a parametrization of a
Roman surface is basepoint free.
The compatible reparametrizations are in this case linear
automorphisms of the projective plane.
'''
y = ring('y0,y1,y2,y3')
x = ring('x0,x1,x2')
c = ring('c0,c1,c2,c3,c4,c5,c6,c7,c8')
# parametrizations f and g of Roman surface
f = ring('[x0^2+x1^2+x2^2,x0*x1,x0*x2,x1*x2]')
g = f
# compatible reparametrizations are linear and indexed by c
r = {
x[0]: c[0] * y[0] + c[1] * y[1] + c[2] * y[2],
x[1]: c[3] * y[0] + c[4] * y[1] + c[5] * y[2],
x[2]: c[6] * y[0] + c[7] * y[1] + c[8] * y[2]
}
# compute kernel and coefficient matrix of f
Mf = SERing.get_matrix_P2(f)
Kf = Mf.right_kernel_matrix().T
assert (Mf * Kf).is_zero()
# compute the coefficient matrix of g composed with r
gr = [comp.subs(r) for comp in g]
assert sage_gcd(gr) == 1
assert SERing.get_degree(gr, 'y0,y1,y2') == SERing.get_degree(f)
Mgr = SERing.get_matrix_P2(gr, 'y0,y1,y2')
# output to screen
SETools.p('f =', f)
SETools.p('g =', g)
SETools.p('r =', r)
SETools.p('Mf =', Mf.dimensions(), list(Mf), SERing.get_mon_P2(2))
SETools.p('Kf.T =', Kf.T.dimensions(), list(Kf.T))
SETools.p('Mgr =', Mgr.dimensions(), list(Mgr), '\n' + str(Mgr))
# compute c such that Mgr*Kf==0
ec_lst = (Mgr * Kf).list() + [
sage_matrix(SERing.R, 3, 3, c).det() * ring('t') - 1
]
pc_lst = sage_ideal(ec_lst).elimination_ideal(
ring('t')).primary_decomposition()
SETools.p('sol_lst = ')
sol_lst = []
for pc in pc_lst:
s_lst = list(reversed(sorted(pc.gens())))
s_dct = ring(
sage_solve([sage_SR(comp) for comp in s_lst],
[sage_SR(comp) for comp in c],
solution_dict=True)[0])
sol_lst += [s_dct]
SETools.p('\t\t', s_lst, '-->', s_dct)
# compute implicit equation for image Y of g
eqg = sage_ideal([y[i] - g[i]
for i in range(4)]).elimination_ideal(x).gens()
SETools.p('eqg =', eqg)
SETools.p(
' =',
str(eqg.subs({y[0]: 1})).replace('y1',
'x').replace('y2',
'y').replace('y3', 'z'))
# computing U and test each sol in sol_lst
SETools.p('Testing each sol in sol_lst...')
for sol in sol_lst:
# compute the projective isomorphism in terms of parametrized matrix U
Ef = sage_matrix(sage_QQ, list(Mf) + list(Kf.T))
Egr = sage_matrix(list(Mgr.subs(sol)) + list(Kf.T))
UpI = Egr * ~Ef
assert (UpI.submatrix(4, 4) - sage_identity_matrix(2)).is_zero()
U = UpI.submatrix(0, 0, 4, 4)
U = U / sage_gcd(U.list())
assert U.dimensions() == (4, 4)
# verify whether U*f is a parametrization for Y for all (c0,...,c7)
Uf = list(U * sage_vector(f))
eqg_sub = [eq.subs({y[i]: Uf[i] for i in range(4)}) for eq in eqg]
assert eqg_sub == [0]
# output U with corresponding solution
SETools.p('\t U =', list(U), ', sol =', sol)
# f is of bidegree (d1, d2) with coefficent matrix matf.
# The image of f is a surface in P^m.
# g is constructed as the composition U o f o P
# where the reparametrization P: P1xP1-->P1xP1 is defined by two 2x2 matrices L and R
# and U is defined by a 4x4 matrix.
d1, d2, m = SERing.random_elt([2, 3]), SERing.random_elt([2, 3]), 3
matf = SERing.random_matrix_QQ(m + 1, len(SERing.get_mon_P1xP1(d1, d2)))
matU = SERing.random_inv_matrix_QQ(m + 1)
L = SERing.random_inv_matrix_QQ(2).list()
R = SERing.random_inv_matrix_QQ(2).list()
if case == 1:
d1, d2, m = 2, 2, 3
matf = sage_matrix(
sage_QQ,
ring(
'[(3/5, 87, -1/2, 1/4, 0, 1/8, 16/5, 0, 0), (0, 11, -1/9, 44, 0, 0, -1, 0, 1/3), (1/12, -4, 0, 2, -1, 1/4, 0, -3, -1/9), (1/3, 1/4, 1, -4, 1/2, -1, -6, 2, 22)]'
))
matU = sage_matrix(
sage_QQ,
ring(
'[( 1, 0, -1, -3 ), ( -1 / 18, -3, 0, 15 / 2 ), ( -1, 0, 13, 1 ), ( -3, -1 / 11, -2, 3 / 23 )]'
))
L = ring('[-1, 0, -1/87, -1/2] ')
R = ring('[2, 1, 0, 2]')
elif case == 2:
d1, d2, m = 3, 3, 3
matf = sage_matrix(
sage_QQ,
ring(