Differentials of function fields

Sage provides arithmetic with differentials of function fields.

EXAMPLES:

The module of differentials on a function field forms an one-dimensional vector space over the function field:

sage: # needs sage.rings.finite_rings sage.rings.function_field
sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x + x^3*Y)
sage: f = x + y
sage: g = 1 / y
sage: df = f.differential()
sage: dg = g.differential()
sage: dfdg = f.derivative() / g.derivative()
sage: df == dfdg * dg
True
sage: df
(x*y^2 + 1/x*y + 1) d(x)
sage: df.parent()
Space of differentials of Function field in y defined by y^3 + x^3*y + x

>>> from sage.all import *
>>> # needs sage.rings.finite_rings sage.rings.function_field
>>> K = FunctionField(GF(Integer(4)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> L = K.extension(Y**Integer(3) + x + x**Integer(3)*Y, names=('y',)); (y,) = L._first_ngens(1)
>>> f = x + y
>>> g = Integer(1) / y
>>> df = f.differential()
>>> dg = g.differential()
>>> dfdg = f.derivative() / g.derivative()
>>> df == dfdg * dg
True
>>> df
(x*y^2 + 1/x*y + 1) d(x)
>>> df.parent()
Space of differentials of Function field in y defined by y^3 + x^3*y + x

# needs sage.rings.finite_rings sage.rings.function_field
K.<x> = FunctionField(GF(4)); _.<Y> = K[]
L.<y> = K.extension(Y^3 + x + x^3*Y)
f = x + y
g = 1 / y
df = f.differential()
dg = g.differential()
dfdg = f.derivative() / g.derivative()
df == dfdg * dg
df
df.parent()

We can compute a canonical divisor:

sage: # needs sage.rings.finite_rings sage.rings.function_field
sage: k = df.divisor()
sage: k.degree()
4
sage: k.degree() == 2 * L.genus() - 2
True

>>> from sage.all import *
>>> # needs sage.rings.finite_rings sage.rings.function_field
>>> k = df.divisor()
>>> k.degree()
4
>>> k.degree() == Integer(2) * L.genus() - Integer(2)
True

# needs sage.rings.finite_rings sage.rings.function_field
k = df.divisor()
k.degree()
k.degree() == 2 * L.genus() - 2

Exact differentials vanish and logarithmic differentials are stable under the Cartier operation:

sage: # needs sage.rings.finite_rings sage.rings.function_field
sage: df.cartier()
0
sage: w = 1/f * df
sage: w.cartier() == w
True

>>> from sage.all import *
>>> # needs sage.rings.finite_rings sage.rings.function_field
>>> df.cartier()
0
>>> w = Integer(1)/f * df
>>> w.cartier() == w
True

# needs sage.rings.finite_rings sage.rings.function_field
df.cartier()
w = 1/f * df
w.cartier() == w

AUTHORS:

  • Kwankyu Lee (2017-04-30): initial version

class sage.rings.function_field.differential.DifferentialsSpace(field, category=None)[source]

Bases: UniqueRepresentation, Parent

Space of differentials of a function field.

INPUT:

  • field – function field

EXAMPLES:

sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[]                                 # needs sage.rings.finite_rings
sage: L.<y> = K.extension(Y^3 + x^3*Y + x)                                      # needs sage.rings.finite_rings sage.rings.function_field
sage: L.space_of_differentials()                                                # needs sage.rings.finite_rings sage.rings.function_field
Space of differentials of Function field in y defined by y^3 + x^3*y + x

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(4)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)# needs sage.rings.finite_rings
>>> L = K.extension(Y**Integer(3) + x**Integer(3)*Y + x, names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.finite_rings sage.rings.function_field
>>> L.space_of_differentials()                                                # needs sage.rings.finite_rings sage.rings.function_field
Space of differentials of Function field in y defined by y^3 + x^3*y + x

K.<x> = FunctionField(GF(4)); _.<Y> = K[]                                 # needs sage.rings.finite_rings
L.<y> = K.extension(Y^3 + x^3*Y + x)                                      # needs sage.rings.finite_rings sage.rings.function_field
L.space_of_differentials()                                                # needs sage.rings.finite_rings sage.rings.function_field

The space of differentials is a one-dimensional module over the function field. So a base differential is chosen to represent differentials. Usually the generator of the base rational function field is a separating element and used to generate the base differential. Otherwise a separating element is automatically found and used to generate the base differential relative to which other differentials are denoted:

sage: # needs sage.rings.function_field
sage: K.<x> = FunctionField(GF(5))
sage: R.<y> = K[]
sage: L.<y> = K.extension(y^5 - 1/x)
sage: L(x).differential()
0
sage: y.differential()
d(y)
sage: (y^2).differential()
(2*y) d(y)

>>> from sage.all import *
>>> # needs sage.rings.function_field
>>> K = FunctionField(GF(Integer(5)), names=('x',)); (x,) = K._first_ngens(1)
>>> R = K['y']; (y,) = R._first_ngens(1)
>>> L = K.extension(y**Integer(5) - Integer(1)/x, names=('y',)); (y,) = L._first_ngens(1)
>>> L(x).differential()
0
>>> y.differential()
d(y)
>>> (y**Integer(2)).differential()
(2*y) d(y)

# needs sage.rings.function_field
K.<x> = FunctionField(GF(5))
R.<y> = K[]
L.<y> = K.extension(y^5 - 1/x)
L(x).differential()
y.differential()
(y^2).differential()
Element[source]

alias of FunctionFieldDifferential

basis()[source]

Return a basis.

EXAMPLES:

sage: # needs sage.rings.finite_rings sage.rings.function_field
sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x^3*Y + x)
sage: S = L.space_of_differentials()
sage: S.basis()
Family (d(x),)

>>> from sage.all import *
>>> # needs sage.rings.finite_rings sage.rings.function_field
>>> K = FunctionField(GF(Integer(4)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> L = K.extension(Y**Integer(3) + x**Integer(3)*Y + x, names=('y',)); (y,) = L._first_ngens(1)
>>> S = L.space_of_differentials()
>>> S.basis()
Family (d(x),)

# needs sage.rings.finite_rings sage.rings.function_field
K.<x> = FunctionField(GF(4)); _.<Y> = K[]
L.<y> = K.extension(Y^3 + x^3*Y + x)
S = L.space_of_differentials()
S.basis()
function_field()[source]

Return the function field to which the space of differentials is attached.

EXAMPLES:

sage: # needs sage.rings.finite_rings sage.rings.function_field
sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x^3*Y + x)
sage: S = L.space_of_differentials()
sage: S.function_field()
Function field in y defined by y^3 + x^3*y + x

>>> from sage.all import *
>>> # needs sage.rings.finite_rings sage.rings.function_field
>>> K = FunctionField(GF(Integer(4)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> L = K.extension(Y**Integer(3) + x**Integer(3)*Y + x, names=('y',)); (y,) = L._first_ngens(1)
>>> S = L.space_of_differentials()
>>> S.function_field()
Function field in y defined by y^3 + x^3*y + x

# needs sage.rings.finite_rings sage.rings.function_field
K.<x> = FunctionField(GF(4)); _.<Y> = K[]
L.<y> = K.extension(Y^3 + x^3*Y + x)
S = L.space_of_differentials()
S.function_field()
class sage.rings.function_field.differential.DifferentialsSpaceInclusion[source]

Bases: Morphism

Inclusion morphisms for extensions of function fields.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)                                    # needs sage.rings.function_field
sage: OK = K.space_of_differentials()
sage: OL = L.space_of_differentials()                                           # needs sage.rings.function_field
sage: OL.coerce_map_from(OK)                                                    # needs sage.rings.function_field
Inclusion morphism:
  From: Space of differentials of Rational function field in x over Rational Field
  To:   Space of differentials of Function field in y defined by y^2 - x*y + 4*x^3

>>> from sage.all import *
>>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1)
>>> L = K.extension(y**Integer(2) - x*y + Integer(4)*x**Integer(3), names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.function_field
>>> OK = K.space_of_differentials()
>>> OL = L.space_of_differentials()                                           # needs sage.rings.function_field
>>> OL.coerce_map_from(OK)                                                    # needs sage.rings.function_field
Inclusion morphism:
  From: Space of differentials of Rational function field in x over Rational Field
  To:   Space of differentials of Function field in y defined by y^2 - x*y + 4*x^3

K.<x> = FunctionField(QQ); R.<y> = K[]
L.<y> = K.extension(y^2 - x*y + 4*x^3)                                    # needs sage.rings.function_field
OK = K.space_of_differentials()
OL = L.space_of_differentials()                                           # needs sage.rings.function_field
OL.coerce_map_from(OK)                                                    # needs sage.rings.function_field
is_injective()[source]

Return True, since the inclusion morphism is injective.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)                                # needs sage.rings.function_field
sage: OK = K.space_of_differentials()
sage: OL = L.space_of_differentials()                                       # needs sage.rings.function_field
sage: OL.coerce_map_from(OK).is_injective()                                 # needs sage.rings.function_field
True

>>> from sage.all import *
>>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1)
>>> L = K.extension(y**Integer(2) - x*y + Integer(4)*x**Integer(3), names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.function_field
>>> OK = K.space_of_differentials()
>>> OL = L.space_of_differentials()                                       # needs sage.rings.function_field
>>> OL.coerce_map_from(OK).is_injective()                                 # needs sage.rings.function_field
True

K.<x> = FunctionField(QQ); R.<y> = K[]
L.<y> = K.extension(y^2 - x*y + 4*x^3)                                # needs sage.rings.function_field
OK = K.space_of_differentials()
OL = L.space_of_differentials()                                       # needs sage.rings.function_field
OL.coerce_map_from(OK).is_injective()                                 # needs sage.rings.function_field
is_surjective()[source]

Return True if the inclusion morphism is surjective.

EXAMPLES:

sage: # needs sage.rings.function_field
sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)
sage: OK = K.space_of_differentials()
sage: OL = L.space_of_differentials()
sage: OL.coerce_map_from(OK).is_surjective()
False
sage: S.<z> = L[]
sage: M.<z> = L.extension(z - 1)
sage: OM = M.space_of_differentials()
sage: OM.coerce_map_from(OL).is_surjective()
True

>>> from sage.all import *
>>> # needs sage.rings.function_field
>>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1)
>>> L = K.extension(y**Integer(2) - x*y + Integer(4)*x**Integer(3), names=('y',)); (y,) = L._first_ngens(1)
>>> OK = K.space_of_differentials()
>>> OL = L.space_of_differentials()
>>> OL.coerce_map_from(OK).is_surjective()
False
>>> S = L['z']; (z,) = S._first_ngens(1)
>>> M = L.extension(z - Integer(1), names=('z',)); (z,) = M._first_ngens(1)
>>> OM = M.space_of_differentials()
>>> OM.coerce_map_from(OL).is_surjective()
True

# needs sage.rings.function_field
K.<x> = FunctionField(QQ); R.<y> = K[]
L.<y> = K.extension(y^2 - x*y + 4*x^3)
OK = K.space_of_differentials()
OL = L.space_of_differentials()
OL.coerce_map_from(OK).is_surjective()
S.<z> = L[]
M.<z> = L.extension(z - 1)
OM = M.space_of_differentials()
OM.coerce_map_from(OL).is_surjective()
class sage.rings.function_field.differential.DifferentialsSpace_global(field, category=None)[source]

Bases: DifferentialsSpace

Space of differentials of a global function field.

INPUT:

  • field – function field

EXAMPLES:

sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[]                                 # needs sage.rings.finite_rings
sage: L.<y> = K.extension(Y^3 + x^3*Y + x)                                      # needs sage.rings.finite_rings sage.rings.function_field
sage: L.space_of_differentials()                                                # needs sage.rings.finite_rings sage.rings.function_field
Space of differentials of Function field in y defined by y^3 + x^3*y + x

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(4)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)# needs sage.rings.finite_rings
>>> L = K.extension(Y**Integer(3) + x**Integer(3)*Y + x, names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.finite_rings sage.rings.function_field
>>> L.space_of_differentials()                                                # needs sage.rings.finite_rings sage.rings.function_field
Space of differentials of Function field in y defined by y^3 + x^3*y + x

K.<x> = FunctionField(GF(4)); _.<Y> = K[]                                 # needs sage.rings.finite_rings
L.<y> = K.extension(Y^3 + x^3*Y + x)                                      # needs sage.rings.finite_rings sage.rings.function_field
L.space_of_differentials()                                                # needs sage.rings.finite_rings sage.rings.function_field
Element[source]

alias of FunctionFieldDifferential_global

class sage.rings.function_field.differential.FunctionFieldDifferential(parent, f, t=None)[source]

Bases: ModuleElement

Base class for differentials on function fields.

INPUT:

  • f – element of the function field

  • t – element of the function field; if \(t\) is not specified, the generator of the base differential is assumed

EXAMPLES:

sage: F.<x> = FunctionField(QQ)
sage: f = x/(x^2 + x + 1)
sage: f.differential()
((-x^2 + 1)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1)) d(x)

>>> from sage.all import *
>>> F = FunctionField(QQ, names=('x',)); (x,) = F._first_ngens(1)
>>> f = x/(x**Integer(2) + x + Integer(1))
>>> f.differential()
((-x^2 + 1)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1)) d(x)

F.<x> = FunctionField(QQ)
f = x/(x^2 + x + 1)
f.differential()
sage: K.<x> = FunctionField(QQ); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x + x^3*Y)                                      # needs sage.rings.function_field
sage: L(x).differential()                                                       # needs sage.rings.function_field
d(x)
sage: y.differential()                                                          # needs sage.rings.function_field
((21/4*x/(x^7 + 27/4))*y^2 + ((3/2*x^7 + 9/4)/(x^8 + 27/4*x))*y + 7/2*x^4/(x^7 + 27/4)) d(x)

>>> from sage.all import *
>>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> L = K.extension(Y**Integer(3) + x + x**Integer(3)*Y, names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.function_field
>>> L(x).differential()                                                       # needs sage.rings.function_field
d(x)
>>> y.differential()                                                          # needs sage.rings.function_field
((21/4*x/(x^7 + 27/4))*y^2 + ((3/2*x^7 + 9/4)/(x^8 + 27/4*x))*y + 7/2*x^4/(x^7 + 27/4)) d(x)

K.<x> = FunctionField(QQ); _.<Y> = K[]
L.<y> = K.extension(Y^3 + x + x^3*Y)                                      # needs sage.rings.function_field
L(x).differential()                                                       # needs sage.rings.function_field
y.differential()                                                          # needs sage.rings.function_field
>>> from sage.all import *
>>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> L = K.extension(Y**Integer(3) + x + x**Integer(3)*Y, names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.function_field
>>> L(x).differential()                                                       # needs sage.rings.function_field
d(x)
>>> y.differential()                                                          # needs sage.rings.function_field
((21/4*x/(x^7 + 27/4))*y^2 + ((3/2*x^7 + 9/4)/(x^8 + 27/4*x))*y + 7/2*x^4/(x^7 + 27/4)) d(x)

K.<x> = FunctionField(QQ); _.<Y> = K[]
L.<y> = K.extension(Y^3 + x + x^3*Y)                                      # needs sage.rings.function_field
L(x).differential()                                                       # needs sage.rings.function_field
y.differential()                                                          # needs sage.rings.function_field
divisor()[source]

Return the divisor of the differential.

EXAMPLES:

sage: K.<x> = FunctionField(GF(5)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x + x^3*Y)                                  # needs sage.rings.function_field
sage: w = (1/y) * y.differential()                                          # needs sage.rings.function_field
sage: w.divisor()                                                           # needs sage.rings.function_field
- Place (1/x, 1/x^3*y^2 + 1/x)
 - Place (1/x, 1/x^3*y^2 + 1/x^2*y + 1)
 - Place (x, y)
 + Place (x + 2, y + 3)
 + Place (x^6 + 3*x^5 + 4*x^4 + 2*x^3 + x^2 + 3*x + 4, y + x^5)

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(5)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> L = K.extension(Y**Integer(3) + x + x**Integer(3)*Y, names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.function_field
>>> w = (Integer(1)/y) * y.differential()                                          # needs sage.rings.function_field
>>> w.divisor()                                                           # needs sage.rings.function_field
- Place (1/x, 1/x^3*y^2 + 1/x)
 - Place (1/x, 1/x^3*y^2 + 1/x^2*y + 1)
 - Place (x, y)
 + Place (x + 2, y + 3)
 + Place (x^6 + 3*x^5 + 4*x^4 + 2*x^3 + x^2 + 3*x + 4, y + x^5)

K.<x> = FunctionField(GF(5)); _.<Y> = K[]
L.<y> = K.extension(Y^3 + x + x^3*Y)                                  # needs sage.rings.function_field
w = (1/y) * y.differential()                                          # needs sage.rings.function_field
w.divisor()                                                           # needs sage.rings.function_field
sage: F.<x> = FunctionField(QQ)
sage: w = (1/x).differential()
sage: w.divisor()                                                           # needs sage.libs.pari
-2*Place (x)

>>> from sage.all import *
>>> F = FunctionField(QQ, names=('x',)); (x,) = F._first_ngens(1)
>>> w = (Integer(1)/x).differential()
>>> w.divisor()                                                           # needs sage.libs.pari
-2*Place (x)

F.<x> = FunctionField(QQ)
w = (1/x).differential()
w.divisor()                                                           # needs sage.libs.pari
>>> from sage.all import *
>>> F = FunctionField(QQ, names=('x',)); (x,) = F._first_ngens(1)
>>> w = (Integer(1)/x).differential()
>>> w.divisor()                                                           # needs sage.libs.pari
-2*Place (x)

F.<x> = FunctionField(QQ)
w = (1/x).differential()
w.divisor()                                                           # needs sage.libs.pari
monomial_coefficients(copy=True)[source]

Return a dictionary whose keys are indices of basis elements in the support of self and whose values are the corresponding coefficients.

EXAMPLES:

sage: K.<x> = FunctionField(GF(5)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x + x^3*Y)                                  # needs sage.rings.function_field
sage: d = y.differential()                                                  # needs sage.rings.function_field
sage: d                                                                     # needs sage.rings.function_field
((4*x/(x^7 + 3))*y^2 + ((4*x^7 + 1)/(x^8 + 3*x))*y + x^4/(x^7 + 3)) d(x)
sage: d.monomial_coefficients()                                             # needs sage.rings.function_field
{0: (4*x/(x^7 + 3))*y^2 + ((4*x^7 + 1)/(x^8 + 3*x))*y + x^4/(x^7 + 3)}

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(5)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> L = K.extension(Y**Integer(3) + x + x**Integer(3)*Y, names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.function_field
>>> d = y.differential()                                                  # needs sage.rings.function_field
>>> d                                                                     # needs sage.rings.function_field
((4*x/(x^7 + 3))*y^2 + ((4*x^7 + 1)/(x^8 + 3*x))*y + x^4/(x^7 + 3)) d(x)
>>> d.monomial_coefficients()                                             # needs sage.rings.function_field
{0: (4*x/(x^7 + 3))*y^2 + ((4*x^7 + 1)/(x^8 + 3*x))*y + x^4/(x^7 + 3)}

K.<x> = FunctionField(GF(5)); _.<Y> = K[]
L.<y> = K.extension(Y^3 + x + x^3*Y)                                  # needs sage.rings.function_field
d = y.differential()                                                  # needs sage.rings.function_field
d                                                                     # needs sage.rings.function_field
d.monomial_coefficients()                                             # needs sage.rings.function_field
residue(place)[source]

Return the residue of the differential at the place.

INPUT:

  • place – a place of the function field

OUTPUT: an element of the residue field of the place

EXAMPLES:

We verify the residue theorem in a rational function field:

sage: # needs sage.rings.finite_rings
sage: F.<x> = FunctionField(GF(4))
sage: f = 0
sage: while f == 0:
....:     f = F.random_element()
sage: w = 1/f * f.differential()
sage: d = f.divisor()
sage: s = d.support()
sage: sum([w.residue(p).trace() for p in s])                                # needs sage.rings.function_field
0

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> F = FunctionField(GF(Integer(4)), names=('x',)); (x,) = F._first_ngens(1)
>>> f = Integer(0)
>>> while f == Integer(0):
...     f = F.random_element()
>>> w = Integer(1)/f * f.differential()
>>> d = f.divisor()
>>> s = d.support()
>>> sum([w.residue(p).trace() for p in s])                                # needs sage.rings.function_field
0

# needs sage.rings.finite_rings
F.<x> = FunctionField(GF(4))
f = 0
while f == 0:
    f = F.random_element()
w = 1/f * f.differential()
d = f.divisor()
s = d.support()
sum([w.residue(p).trace() for p in s])                                # needs sage.rings.function_field

and in an extension field:

sage: # needs sage.rings.function_field
sage: K.<x> = FunctionField(GF(7)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x + x^3*Y)
sage: f = 0
sage: while f == 0:
....:     f = L.random_element()
sage: w = 1/f * f.differential()
sage: d = f.divisor()
sage: s = d.support()
sage: sum([w.residue(p).trace() for p in s])
0

>>> from sage.all import *
>>> # needs sage.rings.function_field
>>> K = FunctionField(GF(Integer(7)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> L = K.extension(Y**Integer(3) + x + x**Integer(3)*Y, names=('y',)); (y,) = L._first_ngens(1)
>>> f = Integer(0)
>>> while f == Integer(0):
...     f = L.random_element()
>>> w = Integer(1)/f * f.differential()
>>> d = f.divisor()
>>> s = d.support()
>>> sum([w.residue(p).trace() for p in s])
0

# needs sage.rings.function_field
K.<x> = FunctionField(GF(7)); _.<Y> = K[]
L.<y> = K.extension(Y^3 + x + x^3*Y)
f = 0
while f == 0:
    f = L.random_element()
w = 1/f * f.differential()
d = f.divisor()
s = d.support()
sum([w.residue(p).trace() for p in s])

and also in a function field of characteristic zero:

sage: # needs sage.rings.function_field
sage: R.<x> = FunctionField(QQ)
sage: L.<Y> = R[]
sage: F.<y> = R.extension(Y^2 - x^4 - 4*x^3 - 2*x^2 - 1)
sage: a = 6*x^2 + 5*x + 7
sage: b = 2*x^6 + 8*x^5 + 3*x^4 - 4*x^3 - 1
sage: w = y*a/b*x.differential()
sage: d = w.divisor()
sage: sum([QQ(w.residue(p)) for p in d.support()])
0

>>> from sage.all import *
>>> # needs sage.rings.function_field
>>> R = FunctionField(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> L = R['Y']; (Y,) = L._first_ngens(1)
>>> F = R.extension(Y**Integer(2) - x**Integer(4) - Integer(4)*x**Integer(3) - Integer(2)*x**Integer(2) - Integer(1), names=('y',)); (y,) = F._first_ngens(1)
>>> a = Integer(6)*x**Integer(2) + Integer(5)*x + Integer(7)
>>> b = Integer(2)*x**Integer(6) + Integer(8)*x**Integer(5) + Integer(3)*x**Integer(4) - Integer(4)*x**Integer(3) - Integer(1)
>>> w = y*a/b*x.differential()
>>> d = w.divisor()
>>> sum([QQ(w.residue(p)) for p in d.support()])
0

# needs sage.rings.function_field
R.<x> = FunctionField(QQ)
L.<Y> = R[]
F.<y> = R.extension(Y^2 - x^4 - 4*x^3 - 2*x^2 - 1)
a = 6*x^2 + 5*x + 7
b = 2*x^6 + 8*x^5 + 3*x^4 - 4*x^3 - 1
w = y*a/b*x.differential()
d = w.divisor()
sum([QQ(w.residue(p)) for p in d.support()])
valuation(place)[source]

Return the valuation of the differential at the place.

INPUT:

  • place – a place of the function field

EXAMPLES:

sage: K.<x> = FunctionField(GF(5)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x + x^3*Y)                                  # needs sage.rings.function_field
sage: w = (1/y) * y.differential()                                          # needs sage.rings.function_field
sage: [w.valuation(p) for p in L.places()]                                  # needs sage.rings.function_field
[-1, -1, -1, 0, 1, 0]

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(5)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> L = K.extension(Y**Integer(3) + x + x**Integer(3)*Y, names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.function_field
>>> w = (Integer(1)/y) * y.differential()                                          # needs sage.rings.function_field
>>> [w.valuation(p) for p in L.places()]                                  # needs sage.rings.function_field
[-1, -1, -1, 0, 1, 0]

K.<x> = FunctionField(GF(5)); _.<Y> = K[]
L.<y> = K.extension(Y^3 + x + x^3*Y)                                  # needs sage.rings.function_field
w = (1/y) * y.differential()                                          # needs sage.rings.function_field
[w.valuation(p) for p in L.places()]                                  # needs sage.rings.function_field
class sage.rings.function_field.differential.FunctionFieldDifferential_global(parent, f, t=None)[source]

Bases: FunctionFieldDifferential

Differentials on global function fields.

EXAMPLES:

sage: F.<x> = FunctionField(GF(7))
sage: f = x/(x^2 + x + 1)
sage: f.differential()
((6*x^2 + 1)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1)) d(x)

>>> from sage.all import *
>>> F = FunctionField(GF(Integer(7)), names=('x',)); (x,) = F._first_ngens(1)
>>> f = x/(x**Integer(2) + x + Integer(1))
>>> f.differential()
((6*x^2 + 1)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1)) d(x)

F.<x> = FunctionField(GF(7))
f = x/(x^2 + x + 1)
f.differential()
sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[]                                 # needs sage.rings.finite_rings
sage: L.<y> = K.extension(Y^3 + x + x^3*Y)                                      # needs sage.rings.finite_rings sage.rings.function_field
sage: y.differential()                                                          # needs sage.rings.finite_rings sage.rings.function_field
(x*y^2 + 1/x*y) d(x)

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(4)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)# needs sage.rings.finite_rings
>>> L = K.extension(Y**Integer(3) + x + x**Integer(3)*Y, names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.finite_rings sage.rings.function_field
>>> y.differential()                                                          # needs sage.rings.finite_rings sage.rings.function_field
(x*y^2 + 1/x*y) d(x)

K.<x> = FunctionField(GF(4)); _.<Y> = K[]                                 # needs sage.rings.finite_rings
L.<y> = K.extension(Y^3 + x + x^3*Y)                                      # needs sage.rings.finite_rings sage.rings.function_field
y.differential()                                                          # needs sage.rings.finite_rings sage.rings.function_field
>>> from sage.all import *
>>> K = FunctionField(GF(Integer(4)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)# needs sage.rings.finite_rings
>>> L = K.extension(Y**Integer(3) + x + x**Integer(3)*Y, names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.finite_rings sage.rings.function_field
>>> y.differential()                                                          # needs sage.rings.finite_rings sage.rings.function_field
(x*y^2 + 1/x*y) d(x)

K.<x> = FunctionField(GF(4)); _.<Y> = K[]                                 # needs sage.rings.finite_rings
L.<y> = K.extension(Y^3 + x + x^3*Y)                                      # needs sage.rings.finite_rings sage.rings.function_field
y.differential()                                                          # needs sage.rings.finite_rings sage.rings.function_field
cartier()[source]

Return the image of the differential by the Cartier operator.

The Cartier operator operates on differentials. Let \(x\) be a separating element of the function field. If a differential \(\omega\) is written in prime-power representation \(\omega=(f_0^p+f_1^px+\dots+f_{p-1}^px^{p-1})dx\), then the Cartier operator maps \(\omega\) to \(f_{p-1}dx\). It is known that this definition does not depend on the choice of \(x\).

The Cartier operator has interesting properties. Notably, the set of exact differentials is precisely the kernel of the Cartier operator and logarithmic differentials are stable under the Cartier operation.

EXAMPLES:

sage: # needs sage.rings.finite_rings sage.rings.function_field
sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x + x^3*Y)
sage: f = x/y
sage: w = 1/f*f.differential()
sage: w.cartier() == w
True

>>> from sage.all import *
>>> # needs sage.rings.finite_rings sage.rings.function_field
>>> K = FunctionField(GF(Integer(4)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)
>>> L = K.extension(Y**Integer(3) + x + x**Integer(3)*Y, names=('y',)); (y,) = L._first_ngens(1)
>>> f = x/y
>>> w = Integer(1)/f*f.differential()
>>> w.cartier() == w
True

# needs sage.rings.finite_rings sage.rings.function_field
K.<x> = FunctionField(GF(4)); _.<Y> = K[]
L.<y> = K.extension(Y^3 + x + x^3*Y)
f = x/y
w = 1/f*f.differential()
w.cartier() == w
sage: # needs sage.rings.finite_rings
sage: F.<x> = FunctionField(GF(4))
sage: f = x/(x^2 + x + 1)
sage: w = 1/f*f.differential()
sage: w.cartier() == w                                                      # needs sage.rings.function_field
True

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> F = FunctionField(GF(Integer(4)), names=('x',)); (x,) = F._first_ngens(1)
>>> f = x/(x**Integer(2) + x + Integer(1))
>>> w = Integer(1)/f*f.differential()
>>> w.cartier() == w                                                      # needs sage.rings.function_field
True

# needs sage.rings.finite_rings
F.<x> = FunctionField(GF(4))
f = x/(x^2 + x + 1)
w = 1/f*f.differential()
w.cartier() == w                                                      # needs sage.rings.function_field
>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> F = FunctionField(GF(Integer(4)), names=('x',)); (x,) = F._first_ngens(1)
>>> f = x/(x**Integer(2) + x + Integer(1))
>>> w = Integer(1)/f*f.differential()
>>> w.cartier() == w                                                      # needs sage.rings.function_field
True

# needs sage.rings.finite_rings
F.<x> = FunctionField(GF(4))
f = x/(x^2 + x + 1)
w = 1/f*f.differential()
w.cartier() == w                                                      # needs sage.rings.function_field