def test__qmat__qpol(self):
x = MARing.x()
q = MARing.q()
g_lst = DSegre.get_ideal_lst()
chk_qpol = 0
for i in range(len(g_lst)):
chk_qpol += q[i] * g_lst[i]
qmat = DSegre.get_qmat()
qpol = list(sage_vector(x).row() * qmat *
sage_vector(x).column())[0][0]
assert qpol == chk_qpol
def test__get_aut_P8__moduli(self):
a, b, c, d, e, f, g, h = ring('a, b, c, d, e, f, g, h')
M = DSegre.get_aut_P8([a, b, c, d, e, f, g, h])
print(M)
x = ring('[x0, x1, x2, x3, x4, x5, x6, x7, x8]')
y = ring('[y0, y1, y2, y3, y4, y5, y6, y7, y8]')
My = M * sage_vector(y)
dct = {x[i]: My[i] for i in range(9)}
pol = ring('x0^2-x1*x2')
print(pol.subs(dct))
# We may use this to check the dimension of the moduli space
# of invariant quadratic forms, since coefficients of some of the
# terms are necessarily 0.
#
chk = ''
chk += '(b^2*c^2*f^2*g^2 - 2*a*b*c*d*f^2*g^2 + a^2*d^2*f^2*g^2 + 2*b^2*c^2*e*f*g*h - 4*a*b*c*d*e*f*g*h + 2*a^2*d^2*e*f*g*h + b^2*c^2*e^2*h^2 - 2*a*b*c*d*e^2*h^2 + a^2*d^2*e^2*h^2 )*y0^2 + '
chk += '(-b^2*c^2*f^2*g^2 + 2*a*b*c*d*f^2*g^2 - a^2*d^2*f^2*g^2 - 2*b^2*c^2*e*f*g*h + 4*a*b*c*d*e*f*g*h - 2*a^2*d^2*e*f*g*h - b^2*c^2*e^2*h^2 + 2*a*b*c*d*e^2*h^2 - a^2*d^2*e^2*h^2)*y1*y2 + '
chk += '(2*b^2*c^2*e*f*g*h - 4*a*b*c*d*e*f*g*h + 2*a^2*d^2*e*f*g*h )*y3*y4 + '
chk += '(-b^2*c^2*e*f*g*h + 2*a*b*c*d*e*f*g*h - a^2*d^2*e*f*g*h )*y5*y6 + '
chk += '(-b^2*c^2*e*f*g*h + 2*a*b*c*d*e*f*g*h - a^2*d^2*e*f*g*h )*y7*y8 + '
chk += '(b^2*c^2*e^2*g^2 - 2*a*b*c*d*e^2*g^2 + a^2*d^2*e^2*g^2 )*y3^2 + '
chk += '(b^2*c^2*f^2*h^2 - 2*a*b*c*d*f^2*h^2 + a^2*d^2*f^2*h^2 )*y4^2 + '
chk += '(2*b^2*c^2*e*f*g^2 - 4*a*b*c*d*e*f*g^2 + 2*a^2*d^2*e*f*g^2 + 2*b^2*c^2*e^2*g*h - 4*a*b*c*d*e^2*g*h + 2*a^2*d^2*e^2*g*h )*y0*y3 + '
chk += '(2*b^2*c^2*f^2*g*h - 4*a*b*c*d*f^2*g*h + 2*a^2*d^2*f^2*g*h + 2*b^2*c^2*e*f*h^2 - 4*a*b*c*d*e*f*h^2 + 2*a^2*d^2*e*f*h^2 )*y0*y4 + '
chk += '(-b^2*c^2*e*f*g^2 + 2*a*b*c*d*e*f*g^2 - a^2*d^2*e*f*g^2 - b^2*c^2*e^2*g*h + 2*a*b*c*d*e^2*g*h - a^2*d^2*e^2*g*h )*y2*y5 + '
chk += '(-b^2*c^2*f^2*g*h + 2*a*b*c*d*f^2*g*h - a^2*d^2*f^2*g*h - b^2*c^2*e*f*h^2 + 2*a*b*c*d*e*f*h^2 - a^2*d^2*e*f*h^2 )*y1*y6 + '
chk += '(-b^2*c^2*f^2*g*h + 2*a*b*c*d*f^2*g*h - a^2*d^2*f^2*g*h - b^2*c^2*e*f*h^2 + 2*a*b*c*d*e*f*h^2 - a^2*d^2*e*f*h^2 )*y2*y7 + '
chk += '(-b^2*c^2*f^2*h^2 + 2*a*b*c*d*f^2*h^2 - a^2*d^2*f^2*h^2 )*y6*y7 + '
chk += '(-b^2*c^2*e*f*g^2 + 2*a*b*c*d*e*f*g^2 - a^2*d^2*e*f*g^2 - b^2*c^2*e^2*g*h + 2*a*b*c*d*e^2*g*h - a^2*d^2*e^2*g*h )*y1*y8 + '
chk += '(-b^2*c^2*e^2*g^2 + 2*a*b*c*d*e^2*g^2 - a^2*d^2*e^2*g^2 )*y5*y8 '
assert pol.subs(dct) == ring(chk)
def get_invariant_qf(c_lst_lst, exc_idx_lst=[]):
'''
Computes quadratic forms in the ideal of the
double Segre surface that are invariant under
a given subgroup of Aut(P^1xP^1).
Parameters
----------
c_lst_lst : list<list<MARing.FF>>
A list of "c_lst"-lists.
A c_lst is a list of length 8 with elements
c0,...,c7 in QQ(k),
where QQ(k) is a subfield of "MARing.FF".
If we substitute k:=0 in the entries of
"c_lst" then we should obtain the list:
[1,0,0,1,1,0,0,1].
A c_lst represents a pair of two matrices:
( [ c0 c1 ] [ c4 c5 ] )
( [ c2 c3 ] , [ c6 c7 ] )
with the property that
c0*c3-c1*c2!=0 and c4*c7-c5*c6!=0.
If the two matrices are not normalized
to have determinant 1 then the method should be
taken with care (it should be checked that the
tangent vectors at the identity generate the
correct Lie algebra).
exc_idx_lst : list<int>
A list of integers in [0,8].
Returns
-------
A list of quadratic forms in the ideal of (a projection of)
the double Segre surface S:
".get_ideal_lst( exc_idx_lst )"
such that the quadratic forms are invariant
under the automorphisms of S as defined by "c_lst_lst"
and such that the quadratic forms generate the module of
all invariant quadratic forms. Note that Aut(S)=Aut(P^1xP^1).
'''
# for verbose output
#
mt = MATools()
sage_set_verbose(-1)
# initialize vectors for indeterminates of "MARing.R"
#
x = MARing.x()
q = MARing.q()
r = MARing.r()
# obtain algebraic conditions on q0,...,q19
# so that the associated quadratic form is invariant
# wrt. the automorphism defined by input "c_lst_lst"
#
iq_lst = []
for c_lst in c_lst_lst:
iq_lst += DSegre.get_invariant_q_lst(c_lst, exc_idx_lst)
iq_lst = list(MARing.R.ideal(iq_lst).groebner_basis())
# solve the ideal defined by "iq_lst"
#
sol_dct = MARing.solve(iq_lst, q)
# substitute the solution in the quadratic form
# associated to the symmetric matrix qmat.
#
qmat = DSegre.get_qmat(exc_idx_lst)
qpol = list(sage_vector(x).row() * qmat *
sage_vector(x).column())[0][0]
sqpol = qpol.subs(sol_dct)
mt.p('sqpol =', sqpol)
mt.p('r =', r)
assert sqpol.subs({ri: 0 for ri in r}) == 0
iqf_lst = [] # iqf=invariant quadratic form
for i in range(len(r)):
coef = sqpol.coefficient(r[i])
if coef != 0:
iqf_lst += [coef]
mt.p('iqf_lst =', iqf_lst)
return iqf_lst
def usecase__horn_and_spindle_cyclides():
'''
We consider the following cyclides with monomial
parametrization determined by the following lattice
polygons:
[ * ] [ * ]
[* * *] [ * ]
[ * ] [* * *]
spindle cyclide horn cyclide
leftright leftright
(2,4,3) (2,4,3)
and 'leftright' involution act as a modular involution
with vertical symmetry axis. We use the following numbering
(see DSegre.get_ideal_lst() and DSegre.change_basis()):
[8 3 5]
[2 0 1]
[6 4 7]
'''
if True:
#
# G-invariant quadratic forms for spindle cyclide
# (see usecase__invariant_quadratic_forms('243ss')):
#
# x0^2 - x3*x4
# x0^2 - x1^2 - x2^2
#
# We send x3 |--> a*(x4-x3)
# x4 |--> a*(x4+x3)
# where a=sqrt(1/2)
# followed by sending: x4 |--> x0, x0 |--> x4
a = ring('a')
xv = x0, x1, x2, x3, x4 = ring('x0,x1,x2,x3,x4')
y0, y1, y2, y3 = ring('y0,y1,y2,y3')
m34 = sage_matrix(MARing.R,
[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, -a, a], [0, 0, 0, a, a]])
m04 = sage_matrix(MARing.R,
[[0, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 0], [1, 0, 0, 0, 0]])
m = m34 * m04
v = sage_vector(xv)
g1 = ring('x0^2 - x3*x4')
g2 = ring('x0^2 - x1^2 - x2^2')
mat1 = sage_invariant_theory.quadratic_form(
g1, xv).as_QuadraticForm().matrix()
mat2 = sage_invariant_theory.quadratic_form(
g2, xv).as_QuadraticForm().matrix()
ng1 = v.row() * m.T * mat1 * m * v.column()
ng2 = v.row() * m.T * mat2 * m * v.column()
G1 = sage_SR(str(ng1[0])).subs({sage_SR('a'): 1 / sage_sqrt(2)})
G2 = sage_SR(str(ng2[0])).subs({sage_SR('a'): 1 / sage_sqrt(2)})
S = sage_PolynomialRing(sage_QQ,
sage_var('x0,x1,x2,x3,x4,y0,y1,y2,y3'))
x0, x1, x2, x3, x4, y0, y1, y2, y3 = S.gens()
G1 = sage__eval(str(G1), S.gens_dict())
G2 = sage__eval(str(G2), S.gens_dict())
assert G1 in S
assert G2 in S
smap = [y0 - (x0 - x3), y1 - x1, y2 - x2, y3 - x4]
prj_lst = S.ideal([G1, G2] + smap).elimination_ideal(
[x0, x1, x2, x3, x4]).gens()
eqn = str(prj_lst[0].subs({y0: 1})).replace('y1', 'x').replace(
'y2', 'y').replace('y3', 'z')
MATools.p(80 * '-')
MATools.p('SPINDLE CYCLIDE')
MATools.p(80 * '-')
MATools.p('We define a projective automorphism m:P^4--->P^4')
MATools.p('\t a = 1/sqrt(2)')
MATools.p('\t m34 =', list(m34))
MATools.p('\t m04 =', list(m04))
MATools.p('\t m = m34 * m04 =', list(m))
MATools.p('\t det(m) =', m.det())
MATools.p('\t v |--> m*v =', v, '|-->', m * v)
MATools.p(
'Generators of ideal of cyclide in quadric of signature [1,4]:')
MATools.p('\t g1 =', g1)
MATools.p('\t g2 =', g2)
MATools.p('Generators of ideal of cyclide in S^3 after applying m:')
MATools.p('\t G1 =', G1)
MATools.p('\t G2 =', G2)
MATools.p('\t G2-2*G1 =', G2 - 2 * G1)
MATools.p('Stereographic projection to circular quadratic cone:')
MATools.p('\t smap =', smap, '(stereographic projection map)')
MATools.p('\t prj_lst =', prj_lst)
MATools.p('\t eqn =', eqn)
MATools.p(80 * '-' + 2 * '\n')
if True:
#
# G-invariant quadratic forms for horn cyclides
# (see usecase__invariant_quadratic_forms('243st')):
#
# x0^2 - x3*x4
# x4^2 - x6^2 - x7^2
#
a = ring('a')
xv = x0, x3, x4, x6, x7 = ring('x0,x3,x4,x6,x7')
y0, y1, y2, y3 = ring('y0,y1,y2,y3')
m = sage_matrix(MARing.R,
[[0, 0, a, 0, 0], [0, 1, 0, 0, 0], [-1, -1, 0, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]])
v = sage_vector(xv)
g1 = ring('x0^2 - x3*x4')
g2 = ring('x4^2 - x6^2 - x7^2')
mat1 = sage_invariant_theory.quadratic_form(
g1, xv).as_QuadraticForm().matrix()
mat2 = sage_invariant_theory.quadratic_form(
g2, xv).as_QuadraticForm().matrix()
ng1 = v.row() * m.T * mat1 * m * v.column()
ng2 = v.row() * m.T * mat2 * m * v.column()
G1 = sage_SR(str(ng1[0])).subs({sage_SR('a'): 1 / sage_sqrt(2)})
G2 = sage_SR(str(ng2[0])).subs({sage_SR('a'): 1 / sage_sqrt(2)})
S = sage_PolynomialRing(sage_QQ,
sage_var('x0,x3,x4,x6,x7,y0,y1,y2,y3'))
x0, x3, x4, x6, x7, y0, y1, y2, y3 = S.gens()
G1 = sage__eval(str(G1), S.gens_dict())
G2 = sage__eval(str(G2), S.gens_dict())
assert G1 in S
assert G2 in S
smap = [y0 - (x0 + x3), y1 - x4, y2 - x7, y3 - x6]
prj_lst = S.ideal([G1, G2] + smap).elimination_ideal(
[x0, x3, x4, x6, x7]).gens()
eqn = str(prj_lst[0].subs({y0: 1})).replace('y1', 'x').replace(
'y2', 'y').replace('y3', 'z')
MATools.p(80 * '-')
MATools.p('HORN CYCLIDE')
MATools.p(80 * '-')
MATools.p('We define a projective automorphism m:P^4--->P^4')
MATools.p('\t a = 1/sqrt(2)')
MATools.p('\t m =', list(m))
MATools.p('\t det(m) =', m.det())
MATools.p('\t v |--> m*v =', v, '|-->', m * v)
MATools.p(
'Generators of ideal of cyclide in quadric of signature [1,4]:')
MATools.p('\t g1 =', g1)
MATools.p('\t g2 =', g2)
MATools.p('Generators of ideal of cyclide in S^3 after applying m:')
MATools.p('\t G1 =', G1)
MATools.p('\t G2 =', G2)
MATools.p('\t G2-2*G1 =', G2 - 2 * G1)
MATools.p('Stereographic projection to circular cylinder:')
MATools.p('\t smap =', smap, '(stereographic projection map)')
MATools.p('\t prj_lst =', prj_lst)
MATools.p('\t eqn =', eqn)
MATools.p(80 * '-' + 2 * '\n')
def get_invariant_qf(c_lst_lst):
'''
Parameters
----------
c_lst_lst : list
A list of "c_lst"-lists.
A c_lst is a list of length 9 with elements
c0,...,c8 in the subring in QQ(k) of "MARing.FF".
The matrix
[ c0 c1 c2 ]
M = [ c3 c4 c5 ]
[ c6 c7 c8 ]
represents---for each value of k---an
automorphism of P^2. If we set k:=0 then
"c_lst" must correspond to the identity matrix:
[ 1,0,0, 0,1,0, 0,0,1 ]. If M is not normalized
to have determinant 1 then the method should be
taken with care (see doc. ".get_c_lst_lst_dct").
Returns
-------
list<MARing.R>
A list of quadratic forms in the ideal of the Veronese surface V
(see ".get_ideal_lst()"), such that the quadratic forms are
invariant under the automorphisms of V as defined by "c_lst_lst"
and such that the quadratic forms generate the module of
all invariant quadratic forms. Note that Aut(V)=Aut(P^2).
'''
# for verbose output
#
mt = MATools()
# initialize vectors for indeterminates of "MARing.R"
#
x = MARing.x()[:6]
q = MARing.q()[:6]
r = MARing.r()[:6]
# obtain algebraic conditions on q0,...,q19
# so that the associated quadratic form is invariant
# wrt. the automorphism defined by input "c_lst_lst"
#
iq_lst = []
for c_lst in c_lst_lst:
iq_lst += Veronese.get_invariant_q_lst(c_lst)
iq_lst = list(MARing.R.ideal(iq_lst).groebner_basis())
# solve the ideal defined by "iq_lst"
#
sol_dct = MARing.solve(iq_lst, q)
# substitute the solution in the quadratic form
# associated to the symmetric matrix qmat.
#
qmat = Veronese.get_qmat()
qpol = list(sage_vector(x).row() * qmat *
sage_vector(x).column())[0][0]
sqpol = qpol.subs(sol_dct)
mt.p('sqpol =', sqpol)
mt.p('r =', r)
assert sqpol.subs({ri: 0 for ri in r}) == 0
iqf_lst = [] # iqf=invariant quadratic form
for i in range(len(r)):
coef = sqpol.coefficient(r[i])
if coef != 0:
iqf_lst += [coef]
mt.p('iqf_lst =', iqf_lst)
return iqf_lst