FiniteDivisorCategory(F, UP, UPUP, R)¶

This category describes finite rational divisors on a curve, that is finite formal sums SUM(n * P) where the n's are integers and the P's are finite rational points on the curve.

0: %

from AbelianMonoid

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

=: (%, %) -> Boolean

from BasicType

~=: (%, %) -> Boolean

from BasicType

coerce: % -> OutputForm
decompose: % -> Record(id: FractionalIdeal(UP, Fraction UP, UPUP, R), principalPart: R)

decompose(d) returns [id, f] where d = (id) + div(f).

divisor: (F, F) -> %

divisor(a, b) makes the divisor P: (x = a, y = b). Error: if P is singular.

divisor: (F, F, Integer) -> %

divisor(a, b, n) makes the divisor nP where P: (x = a, y = b). P is allowed to be singular if n is a multiple of the rank.

divisor: (R, UP, UP) -> %

divisor(h, d, g) returns gcd of divisor of zeros of h and divisor of zeros of d. d must be squarefree. All ramified zeros of d must be contained in zeros of g.

divisor: (R, UP, UP, UP, F) -> %

divisor(h, d, d', g, r) returns the sum of all the finite points where h/d has residue r. h must be integral. d must be squarefree. d' is some derivative of d (not necessarily dd/dx). g = gcd(d, discriminant) contains the ramified zeros of d

divisor: FractionalIdeal(UP, Fraction UP, UPUP, R) -> %

divisor(I) makes a divisor D from an ideal I.

divisor: R -> %

divisor(g) returns the divisor of the function g.

generator: % -> Union(R, failed)

generator(d) returns f if (f) = d, “failed” if d is not principal. d is assumed to be of degree 0.

generator: (%, Integer, List UP) -> Union(R, failed)

generator(d, k, lp) returns f if (f) = d, “failed” if d is not principal. k is sum of orders of d at special places. Special places are places over infinity and over zeros of polynomials in lp. Elements of lp are assumed to be relatively prime.

hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

ideal: % -> FractionalIdeal(UP, Fraction UP, UPUP, R)

ideal(D) returns the ideal corresponding to a divisor D.

latex: % -> String

from SetCategory

opposite?: (%, %) -> Boolean

from AbelianMonoid

principal?: % -> Boolean

principal?(D) tests if the argument is the divisor of a function.

reduce: % -> %

reduce(D) converts D to some reduced form (the reduced forms can be different in different implementations).

sample: %

from AbelianMonoid

subtractIfCan: (%, %) -> Union(%, failed)
zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

BasicType

CancellationAbelianMonoid

SetCategory