Spaces of differentials of function fields can be created and the differentials belonging to them manipulated. Divisors can be created from differentials and modules generated by a collection of differentials can be formed.
The space of differentials of the algebraic function field F/k.
The simplest ways of creating a differential are given below.
Create the differential d(a) of the function field or order element a.
The identity differential of the space of differentials D.
Returns true iff the divisor D is canonical and a differential having D as its divisor.
The function field of the differentials in the space D.
The function field of the differential d.
A vector space and the isomorphism from this space to the space of differentials of the first kind (holomorphic differentials) of the function field F/k.
A basis of the space of differentials of the first kind (holomorphic differentials) of the function field F/k.
Computes a basis for the space of differentials Ω(D) := { ω ∈Ω(F/k) | (ω) ≥D } for a divisor D of an algebraic function field F/k.
A vector space and the isomorphism from this space to the differential space of the divisor D.
> Q := Rationals();
> Qx<x> := PolynomialRing(Q);
> Qxy<y> := PolynomialRing(Qx);
> f1 := y^2 - (x-1)*(x-2)*(x-3)*(x-5)*(x-6);
> F := FunctionField(f1);
> d := Divisor(F.1) + Divisor(F!BaseRing(F).1);
> V1 := DifferentialSpace(d);
> d := 2*Divisor(F.1) - Divisor(F!BaseRing(F).1);
> V2 := DifferentialSpace(d);
> V1;
KModule V1 of dimension 2 over Rational Field
> V2;
KModule V2 of dimension 2 over Rational Field
> V1 meet V2;
KModule of dimension 0 over Rational Field
> D := DifferentialSpace(F);
> v := V1 ! [2/9, 4/9]; v;
V1: (2/9 4/9)
> D!v;
(2/9*x^2 + 4/9*x) d(x)
> V1!$1;
V1: (2/9 4/9)
> BasisOfDifferentialsFirstKind(F);
[ (x/(x^5 - 17*x^4 + 107*x^3 - 307*x^2 + 396*x - 180)*F.1) d(x), (1/(x^5 -
17*x^4 + 107*x^3 - 307*x^2 + 396*x - 180)*F.1) d(x) ]
Return true if the spaces of differentials D1 and D2 are the same.
A number of general operations for elements are also provided for differentials as well as a number of specific functions for differentials.
The operations on differentials are inherited from the vector space structure of the space of differentials. Additionally, this space is one--dimensional as a vector space over the function field itself. The quotient x/y of two differentials x and y then gives the unique r ∈F such that x = ry.
Returns true if x and y are the same differential.
Returns true if x is in the space of differentials D.
Return whether d is known to be an exact differential. If true additionally return a generator. If d is not already known to be exact then no attempts to determine whether d is exact or not are currently undertaken.
Return true if d is the zero differential.
The valuation of the differential d at the place P.
The divisor (d) of the differential d.
The residue of the differential d at the place P, where P must (currently) have degree one.
> PF<x> := PolynomialRing(GF(31, 3)); > P<y> := PolynomialRing(PF); > FF1<b> := ext<FieldOfFractions(PF) | y^2 - x>; > P<y> := PolynomialRing(FF1); > FF2<d> := ext<FF1 | y^3 - b : Check := false>; > Differential(d); (26/x*d) d(x) > I := Random(FF2, 2)*MaximalOrderInfinite(FF2); > P := Place(Factorization(I)[1][1]); > Valuation(Differential(d), P); -2 > IsExact(Differential(Random(FF2, 2))); true
IsBasis: BoolElt Default: false
PreImages: BoolElt Default: false
The R-module generated by the differentials in the sequence L as an abstract module, together with the map into the space of differentials. The resulting modules can be used for intersection and inner sum computations.If the optional parameter IsBasis is set true the function assumes that the given elements form a basis of the module to be computed.
If the optional parameter PreImages is set true then the preimages of the given elements under the map are returned as the third return value.
Both optional parameters are mainly used to save computation time.
The module of R-linear relations between the differentials of the sequence L. The integer m is used for the following: Let the elements of L be a1, ..., an, V be the relation module ⊆Rn and define M := { ∑i=1m vi ai | v = (vi)i ∈V }. The function tries to compute a generating system of V such that the corresponding generating system of M consists of "small" elements.
> Q := Rationals();
> Qx<x> := PolynomialRing(Q);
> Qxy<y> := PolynomialRing(Qx);
> f1 := y^2 - (x-1)*(x-2)*(x-3)*(x-5)*(x-6);
> F := FunctionField(f1);
> D := DifferentialSpace(F);
> M7 := Module([Differential(3*F.1)], FieldOfFractions(Qx));
> M8 := Module([Differential(F.1), Differential(F!BaseRing(F).1)],
> FieldOfFractions(Qx));
> M12 := M7 meet M8;
> M16 := M7 + M8;
> assert M12 subset M7;
> assert M12 subset M8;
> assert M12 subset M16;
> r := M7![&+[Random([-100, 100])/Random([1, 100])*x^i : i in [1 .. 5]]/
> &+[Random([-100, 100])/Random([1,
> 100])*x^i : i in [1 .. 5]] : j in [1 .. Dimension(M7)]];
> assert M7!D!r eq r;
> r;
M7: ((1/100*x^4 + x^3 - x^2 + 1/100*x - 1/100)/(x^4 - 1/100*x^3 - 1/100*x^2 +
1/100*x - 1/100))
> D!r;
((3/40*x^8 + 162/25*x^7 - 20937/200*x^6 + 114873/200*x^5 - 279531/200*x^4 +
304167/200*x^3 - 24321/40*x^2 + 303/20*x - 297/50)/(x^9 - 1701/100*x^8 +
2679/25*x^7 - 30789/100*x^6 + 19891/50*x^5 - 3593/20*x^4 - 63/10*x^3 +
883/100*x^2 - 144/25*x + 9/5)*F.1) d(x)
The result of applying the Cartier operator r times to b. More precisely, let F/k be a function field over the perfect field k, x ∈F be a separating variable and b = g dx ∈Ω(F/k) with g ∈F be a differential. The Cartier operator is defined by C(b) = ( - dp - 1 g / dxp - 1 )1/p dx. This function computes the r-th iterated application of C to b.
Compute a row representation matrix of the Cartier operator on a basis of the space of holomorphic differentials of F/k (applied r times). More precisely, let F/k be a function field over the perfect field k, ω1, ..., ωg ∈Ω(F/k) be a basis for the holomorphic differentials and r ∈Z≥1. Let M = (λi, j)i, j ∈kg x g be the matrix such that Cr(ωi) = ∑m=1g λi, m ωm for all 1 ≤i ≤g. This function returns M and (ω1, ..., ωg).
> PF<x> := PolynomialRing(GF(31, 3)); > P<y> := PolynomialRing(PF); > FF1<b> := ext<FieldOfFractions(PF) | y^2 - x>; > P<y> := PolynomialRing(FF1); > FF2<d> := ext<FF1 | y^3 - b : Check := false>; > Cartier(Differential(d), 4); (0) d(x) > CartierRepresentation(FF2, 3); Matrix with 0 rows and 0 columns []