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 get_qmat(exc_idx_lst=[]):
'''
Parameters
----------
exc_idx_lst : list<int>
A list of integers in [0,8].
Returns
-------
sage_matrix<MARing.R>
A symmetric 9x9 matrix with entries
in the ring QQ[q0,...,q19] which is a subring
of "MARing.R". It represents the Gramm matrix
of a quadratic form in the ideal of the
double Segre surface or a projection of
the double Segre surface with ideal defined
by "get_ideal_lst( exc_idx_lst )".
Method
------
We obtain generators of the ideal
of the (projection of) the double Segre surface
with the method "get_ideal_lst( exc_idx_lst )".
If the ideal is of a projection of the double Segre
surface, then the returned matrix with parameters
q0,...,q19 is not of full rank.
'''
x = MARing.x()
q = MARing.q()
g_lst = DSegre.get_ideal_lst(exc_idx_lst)
qpol = 0
for i in range(len(g_lst)):
qpol += q[i] * g_lst[i]
qmat = sage_invariant_theory.quadratic_form(
qpol, x).as_QuadraticForm().matrix()
qmat = sage_matrix(MARing.R, qmat)
return qmat
def get_qmat():
'''
Returns
-------
sage_matrix
A symmetric 5x5 matrix with entries in the
ring QQ[q0,...,q5] which is a subring
of "MARing.R". It represents the Gramm matrix
of a quadratic form in the ideal of the
Veronese with ideal defined by ".get_ideal_lst()".
'''
x = MARing.x()[:6]
q = MARing.q()[:6]
g_lst = Veronese.get_ideal_lst()
qpol = 0
for i in range(len(g_lst)):
qpol += q[i] * g_lst[i]
qmat = sage_invariant_theory.quadratic_form(
qpol, x).as_QuadraticForm().matrix()
qmat = sage_matrix(MARing.R, qmat)
return qmat
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 change_basis(iqf_lst, involution='identity'):
'''
This method allows us to change coordinates of the
ideal generated by "iqf_lst" so that
the antiholomorphic involution defined by "involution"
becomes complex conjugation.
Parameters
----------
iqf_lst: list<MARing>
A list of elements in the subring NF[x0,...,x8]
of "MARing.R" where NF denotes the Gaussian
rationals QQ(I) with I^2=-1.
involution: str
Either one of the following strings:
'identity', 'leftright', 'rotate', 'diagonal'.
Returns
-------
We consider the input "iqf_lst" as a map.
We compose this map composed with the map corresponding to
<identity>, <leftright>, <rotate> or <diagonal>.
We return a list of elements in NF[x0,...,x8] that
represents the composition.
Notes
-----
A double Segre surface in projective 8-space P^8 is represented
by a toric parametrization whose completion to P^1xP^1 is provided
by "get_pmz_lst":
(s,u) |-->
(1:s:s^{-1}:u:u^{-1}:s*u:s^{-1}*u^{-1}:s*u^{-1}:s^{-1}*u)
=
(x0:x1:x2:x3:x4:x5:x6:x7:x8)
We can put the exponents of the monomials in a lattice
where x0 corresponds to coordinate (0,0), x6 to (-1,-1)
x5 to (1,1) and x8 to (-1,1):
x8 x3 x5
x2 x0 x1
x6 x4 x7
An antiholomorphic involution, that preserves the toric structure,
acts on the above lattice as a unimodular involution:
identity_*: ( a, b ) |--> ( a, b)
leftright_*: ( a, b ) |--> (-a, b)
rotate_*: ( a, b ) |--> (-a,-b)
diagonal_*: ( a, b ) |--> ( b, a)
These unimodular lattice involutions induce an involution on P^8.
We compose the toric parametrization of the double Segre surface
with the one of the following maps in order to make the antiholomorphic
involution that acts on P^8, equal to complex conjugation.
identity: (x0:...:x8) |--> (x0:...:x8)
leftright: x3 |--> x3,
x0 |--> x0,
x4 |--> x4,
x1 |--> x1 + I*x2,
x2 |--> x1 - I*x2,
x5 |--> x5 + I*x8,
x8 |--> x5 - I*x8,
x7 |--> x7 + I*x6,
x6 |--> x7 - I*x6
rotate: x0 |--> x0,
x1 |--> x1 + I*x2, x2 |--> x1 - I*x2,
x3 |--> x3 + I*x4, x4 |--> x3 - I*x4,
x5 |--> x5 + I*x6, x6 |--> x5 - I*x6,
x7 |--> x7 + I*x8, x8 |--> x7 - I*x8
diagonal: x5 |--> x5,
x0 |--> x0,
x6 |--> x6,
x3 |--> x3 + I*x1,
x1 |--> x3 - I*x1,
x8 |--> x8 + I*x7,
x7 |--> x8 - I*x7,
x2 |--> x2 + I*x4,
x4 |--> x2 - I*x4
'''
I = ring('I')
x = x0, x1, x2, x3, x4, x5, x6, x7, x8 = MARing.x()
z = z0, z1, z2, z3, z4, z5, z6, z7, z8 = MARing.z()
dct = {}
dct['identity'] = {x[i]: z[i] for i in range(9)}
dct['rotate'] = {
x0: z0,
x1: z1 + I * z2,
x2: z1 - I * z2,
x3: z3 + I * z4,
x4: z3 - I * z4,
x5: z5 + I * z6,
x6: z5 - I * z6,
x7: z7 + I * z8,
x8: z7 - I * z8
}
dct['leftright'] = {
x0: z0,
x3: z3,
x4: z4,
x5: z5 + I * z8,
x8: z5 - I * z8,
x1: z1 + I * z2,
x2: z1 - I * z2,
x7: z7 + I * z6,
x6: z7 - I * z6
}
dct['diagonal'] = {
x0: z0,
x6: z6,
x5: z5,
x3: z3 + I * z1,
x1: z3 - I * z1,
x8: z8 + I * z7,
x7: z8 - I * z7,
x2: z2 + I * z4,
x4: z2 - I * z4
}
zx_dct = {z[i]: x[i] for i in range(9)}
new_lst = [iqf.subs(dct[involution]).subs(zx_dct) for iqf in iqf_lst]
return new_lst
def change_basis(iqf_lst):
'''
Parameters
----------
iqf_lst : list
A list of elements in the subring NF[x0,...,x5]
of "MARing.R" where NF denotes the Gaussian
rationals QQ(I) with I^2=-1.
Returns
-------
list
The input "iqf_lst" where we make the following substitution
for each polynomial
x0 |--> x0,
x1 |--> x1,
x2 |--> x2 + I*x3,
x3 |--> x2 - I*x3,
x4 |--> x4 + I*x5,
x5 |--> x4 - I*x5.
Notes
-----
We consider a toric parametrization of the Veronese surface:
(s,t)
|-->
(1 : s*t : s : t : s^2 : t^2 )
=
(x0 : x1 : x2 : x3 : x4 : x5 )
We can put the exponents of the monomials in a lattice
where x0 corresponds to coordinate (0,0), x4 to (2,0)
and x5 to (0,2):
x5 x7 x8
x3 x1 x6
x0 x2 x4
The monomial parametrization of the Veronese corresponds to the
following lattice polygon:
*
* *
* * *
An antiholomorphic involution acts on the above lattice polygon
as a unimodular involution:
( a, b ) |--> ( b, a)
This unimodular lattice involutions induce an involution on P^5:
x0 |--> x0,
x1 |--> x1,
x2 |--> x2 + I*x3,
x3 |--> x2 - I*x3,
x4 |--> x4 + I*x5,
x5 |--> x4 - I*x5,
'''
I = ring('I')
x = x0, x1, x2, x3, x4, x5 = MARing.x()[:6]
z = z0, z1, z2, z3, z4, z5 = MARing.z()[:6]
dct = {
x0: z0,
x1: z1,
x2: z2 + I * z3,
x3: z2 - I * z3,
x4: z4 + I * z5,
x5: z4 - I * z5
}
zx_dct = {z[i]: x[i] for i in range(6)}
new_lst = [iqf.subs(dct).subs(zx_dct) for iqf in iqf_lst]
return new_lst
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
def test__get_aut_P8__rotation_matrix(self):
'''
OUTPUT:
- We verify that for the 1-parameter subgroup of rotations
with matrix
[ cos(k) -sin(k) ]
[ sin(k) cos(k) ]
it is for our Lie algebra methods sufficient to consider
1-parameter subgroups that have the same tangent vector
at the identity. Note that for k=0 we need to get the identity
and the determinant should be 1.
'''
k = ring('k')
I = ring('I')
a, b, c, d, e, f, g, h = ring('a, b, c, d, e, f, g, h')
x = MARing.x()
q = MARing.q()
r = MARing.r()
#
# We consider the representation of the
# following matrix into P^8
#
# ( [ 1 -k ] , [ 1 0 ] )
# ( [ k 1 ] [ 0 1 ] )
#
# We define A to be the tangent vector at the
# identity of this representation
#
N = DSegre.get_aut_P8([1, -k, k, 1] + [1, 0, 0, 1])
A = MARing.diff_mat(N, k).subs({k: 0})
#
# We consider the representation of the
# following matrix into P^8
#
# ( [ cos(k) -sin(k) ] , [ 1 0 ] )
# ( [ sin(k) cos(k) ] [ 0 1 ] )
#
# We define B to be the tangent vector at the
# identity of this representation
#
M = DSegre.get_aut_P8([a, b, c, d] + [e, f, g, h])
a, b, c, d, e, f, g, h = sage_var('a, b, c, d, e, f, g, h')
k = sage_var('k')
M = sage__eval(str(list(M)), {
'a': a,
'b': b,
'c': c,
'd': d,
'e': e,
'f': f,
'g': g,
'h': h,
'k': k
})
M = sage_matrix(M)
M = M.subs({
a: sage_cos(k),
b: -sage_sin(k),
c: sage_sin(k),
d: sage_cos(k),
e: 1,
f: 0,
g: 0,
h: 1
})
# differentiate the entries of M wrt. k
dmat = []
for row in M:
drow = []
for col in row:
drow += [sage_diff(col, k)]
dmat += [drow]
M = sage_matrix(dmat)
B = M.subs({k: 0})
assert str(A) == str(B)