def _linear_system_as_kernel(self, d, pt, m):
"""
Return a matrix whose kernel consists of the coefficient vectors
of the degree d hypersurfaces (wrt lexicographic ordering of its
monomials) with multiplicity at least m at pt.
INPUT:
- ``d`` -- a nonnegative integer
- ``pt`` -- a point of self (possibly represented by a list with at \
least one component equal to 1)
- ``m`` -- a nonnegative integer
OUTPUT:
A matrix of size `{m-1+n \choose n}` x `{d+n \choose n}` where n is the
relative dimension of self. The base ring of the matrix is a ring that
contains the base ring of self and the coefficients of the given point.
EXAMPLES:
If the degree `d` is 0, then a matrix consisting of the first unit vector
is returned::
sage: P = ProjectiveSpace(GF(5), 2, names='x')
sage: pt = P([1, 1, 1])
sage: P._linear_system_as_kernel(0, pt, 3)
[1]
[0]
[0]
[0]
[0]
[0]
If the multiplcity `m` is 0, then the a matrix with zero rows is returned::
sage: P = ProjectiveSpace(GF(5), 2, names='x')
sage: pt = P([1, 1, 1])
sage: M = P._linear_system_as_kernel(2, pt, 0)
sage: [M.nrows(), M.ncols()]
[0, 6]
The base ring does not need to be a field or even an integral domain.
In this case, the point can be given by a list::
sage: R = Zmod(4)
sage: P = ProjectiveSpace(R, 2, names='x')
sage: pt = [R(1), R(3), R(0)]
sage: P._linear_system_as_kernel(3, pt, 2)
[1 3 0 1 0 0 3 0 0 0]
[0 1 0 2 0 0 3 0 0 0]
[0 0 1 0 3 0 0 1 0 0]
When representing a point by a list at least one component must be 1
(even when the base ring is a field and the list gives a well-defined
point in projective space)::
sage: R = GF(5)
sage: P = ProjectiveSpace(R, 2, names='x')
sage: pt = [R(3), R(3), R(0)]
sage: P._linear_system_as_kernel(3, pt, 2)
Traceback (most recent call last):
...
TypeError: At least one component of pt=[3, 3, 0] must be equal
to 1
The components of the list do not have to be elements of the base ring
of the projective space. It suffices if there exists a common parent.
For example, the kernel of the following matrix corresponds to
hypersurfaces of degree 2 in 3-space with multiplicity at least 2 at a
general point in the third affine patch::
sage: P = ProjectiveSpace(QQ,3,names='x')
sage: RPol.<t0,t1,t2,t3> = PolynomialRing(QQ,4)
sage: pt = [t0,t1,1,t3]
sage: P._linear_system_as_kernel(2,pt,2)
[ 2*t0 t1 1 t3 0 0 0 0 0 0]
[ 0 t0 0 0 2*t1 1 t3 0 0 0]
[ t0^2 t0*t1 t0 t0*t3 t1^2 t1 t1*t3 1 t3 t3^2]
[ 0 0 0 t0 0 0 t1 0 1 2*t3]
.. TODO::
Use this method as starting point to implement a class
LinearSystem for linear systems of hypersurfaces.
"""
if not isinstance(d, (int, Integer)):
raise TypeError('The argument d=%s must be an integer' % d)
if d < 0:
raise ValueError('The integer d=%s must be nonnegative' % d)
if not isinstance(pt, (list, tuple, \
SchemeMorphism_point_projective_ring)):
raise TypeError('The argument pt=%s must be a list, tuple, or '
'point on a projective space' % pt)
pt, R = prepare(pt, None)
n = self.dimension_relative()
if not len(pt) == n + 1:
raise TypeError('The sequence pt=%s must have %s '
'components' % (pt, n + 1))
if not R.has_coerce_map_from(self.base_ring()):
raise TypeError('Unable to find a common ring for all elements')
try:
i = pt.index(1)
except Exception:
raise TypeError('At least one component of pt=%s must be equal '
'to 1' % pt)
pt = pt[:i] + pt[i + 1:]
if not isinstance(m, (int, Integer)):
raise TypeError('The argument m=%s must be an integer' % m)
if m < 0:
raise ValueError('The integer m=%s must be nonnegative' % m)
# the components of partials correspond to partial derivatives
# of order at most m-1 with respect to n variables
partials = IntegerVectors(m - 1, n + 1).list()
# the components of monoms correspond to monomials of degree
# at most d in n variables
monoms = IntegerVectors(d, n + 1).list()
M = matrix(R, len(partials), len(monoms))
for row in range(M.nrows()):
e = partials[row][:i] + partials[row][i + 1:]
for col in range(M.ncols()):
f = monoms[col][:i] + monoms[col][i + 1:]
if min([f[j] - e[j] for j in range(n)]) >= 0:
M[row,col] = prod([binomial(f[j],e[j])*pt[j]**(f[j]-e[j]) \
for j in filter(lambda k: f[k]>e[k], range(n))])
return M
def _linear_system_as_kernel(self, d, pt, m):
"""
Return a matrix whose kernel consists of the coefficient vectors
of the degree d hypersurfaces (wrt lexicographic ordering of its
monomials) with multiplicity at least m at pt.
INPUT:
- ``d`` -- a nonnegative integer
- ``pt`` -- a point of self (possibly represented by a list with at \
least one component equal to 1)
- ``m`` -- a nonnegative integer
OUTPUT:
A matrix of size `{m-1+n \choose n}` x `{d+n \choose n}` where n is the
relative dimension of self. The base ring of the matrix is a ring that
contains the base ring of self and the coefficients of the given point.
EXAMPLES:
If the degree `d` is 0, then a matrix consisting of the first unit vector
is returned::
sage: P = ProjectiveSpace(GF(5), 2, names='x')
sage: pt = P([1, 1, 1])
sage: P._linear_system_as_kernel(0, pt, 3)
[1]
[0]
[0]
[0]
[0]
[0]
If the multiplcity `m` is 0, then the a matrix with zero rows is returned::
sage: P = ProjectiveSpace(GF(5), 2, names='x')
sage: pt = P([1, 1, 1])
sage: M = P._linear_system_as_kernel(2, pt, 0)
sage: [M.nrows(), M.ncols()]
[0, 6]
The base ring does not need to be a field or even an integral domain.
In this case, the point can be given by a list::
sage: R = Zmod(4)
sage: P = ProjectiveSpace(R, 2, names='x')
sage: pt = [R(1), R(3), R(0)]
sage: P._linear_system_as_kernel(3, pt, 2)
[1 3 0 1 0 0 3 0 0 0]
[0 1 0 2 0 0 3 0 0 0]
[0 0 1 0 3 0 0 1 0 0]
When representing a point by a list at least one component must be 1
(even when the base ring is a field and the list gives a well-defined
point in projective space)::
sage: R = GF(5)
sage: P = ProjectiveSpace(R, 2, names='x')
sage: pt = [R(3), R(3), R(0)]
sage: P._linear_system_as_kernel(3, pt, 2)
Traceback (most recent call last):
...
TypeError: At least one component of pt=[3, 3, 0] must be equal
to 1
The components of the list do not have to be elements of the base ring
of the projective space. It suffices if there exists a common parent.
For example, the kernel of the following matrix corresponds to
hypersurfaces of degree 2 in 3-space with multiplicity at least 2 at a
general point in the third affine patch::
sage: P = ProjectiveSpace(QQ,3,names='x')
sage: RPol.<t0,t1,t2,t3> = PolynomialRing(QQ,4)
sage: pt = [t0,t1,1,t3]
sage: P._linear_system_as_kernel(2,pt,2)
[ 2*t0 t1 1 t3 0 0 0 0 0 0]
[ 0 t0 0 0 2*t1 1 t3 0 0 0]
[ t0^2 t0*t1 t0 t0*t3 t1^2 t1 t1*t3 1 t3 t3^2]
[ 0 0 0 t0 0 0 t1 0 1 2*t3]
.. TODO::
Use this method as starting point to implement a class
LinearSystem for linear systems of hypersurfaces.
"""
if not isinstance(d, (int, Integer)):
raise TypeError('The argument d=%s must be an integer'%d)
if d < 0:
raise ValueError('The integer d=%s must be nonnegative'%d)
if not isinstance(pt, (list, tuple, \
SchemeMorphism_point_projective_ring)):
raise TypeError('The argument pt=%s must be a list, tuple, or '
'point on a projective space'%pt)
pt, R = prepare(pt, None)
n = self.dimension_relative()
if not len(pt) == n+1:
raise TypeError('The sequence pt=%s must have %s '
'components'%(pt, n + 1))
if not R.has_coerce_map_from(self.base_ring()):
raise TypeError('Unable to find a common ring for all elements')
try:
i = pt.index(1)
except Exception:
raise TypeError('At least one component of pt=%s must be equal '
'to 1'%pt)
pt = pt[:i] + pt[i+1:]
if not isinstance(m, (int, Integer)):
raise TypeError('The argument m=%s must be an integer'%m)
if m < 0:
raise ValueError('The integer m=%s must be nonnegative'%m)
# the components of partials correspond to partial derivatives
# of order at most m-1 with respect to n variables
partials = IntegerVectors(m-1,n+1).list()
# the components of monoms correspond to monomials of degree
# at most d in n variables
monoms = IntegerVectors(d,n+1).list()
M = matrix(R,len(partials),len(monoms))
for row in range(M.nrows()):
e = partials[row][:i] + partials[row][i+1:]
for col in range(M.ncols()):
f = monoms[col][:i] + monoms[col][i+1:]
if min([f[j]-e[j] for j in range(n)]) >= 0:
M[row,col] = prod([ binomial(f[j],e[j]) * pt[j]**(f[j]-e[j])
for j in (k for k in range(n) if f[k] > e[k]) ])
return M