# Differentials 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. ## Creation of Structures ### `DifferentialSpace(F): FldFunG -> DiffFun` The space of differentials of the algebraic function field $F/k$. ## Creation of Elements The simplest ways of creating a differential are given below. ### `Differential(a): FldFunGElt -> DiffFunElt` ### `Differential(a): RngFunOrdElt -> DiffFunElt` Create the differential $d(a)$ of the function field or order element $a$. ### `Identity(D): DiffFun -> DiffFunElt` ### `Id(D): DiffFun -> DiffFunElt` The identity differential of the space of differentials $D$. ### `IsCanonical(D): DivFunElt -> BoolElt, DiffFunElt` Returns `true` iff the divisor $D$ is canonical and a differential having $D$ as its divisor. ## Related Structures ### `FunctionField(D): DiffFun -> FldFun` The function field of the differentials in the space $D$. ### `FunctionField(d): DiffFunElt -> FldFun` The function field of the differential $d$. ## Subspaces ### `SpaceOfDifferentialsFirstKind(F): FldFunG -> ModFld, Map` ### `SpaceOfHolomorphicDifferentials(F): FldFunG -> ModFld, Map` A vector space and the isomorphism from this space to the space of differentials of the first kind (holomorphic differentials) of the function field $F/k$. ### `BasisOfDifferentialsFirstKind(F): FldFunG -> SeqEnum[DiffFunElt]` ### `BasisOfHolomorphicDifferentials(F): FldFunG -> SeqEnum[DiffFunElt]` A basis of the space of differentials of the first kind (holomorphic differentials) of the function field $F/k$. ### `DifferentialBasis(D): DivFunElt -> [DiffFunElt]` Computes a basis for the space of differentials $$ \Omega(D) := \{\; \omega \in \Omega(F/k) \;|\; (\omega) \geq D \;\} $$ for a divisor $D$ of an algebraic function field $F/k$. ### `DifferentialSpace(D): DivFunElt -> ModFld, Map` A vector space and the isomorphism from this space to the differential space of the divisor $D$. ### `Example: Div Diff (ex-b69721)` This example illustrates the differential space of a divisor and some of the operations that can be done with it. ```magma > Q := Rationals(); > Qx := PolynomialRing(Q); > Qxy := 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) ] ``` ## Structure Predicates ### `D1 eq D2: DiffFun, DiffFun -> BoolElt` Return `true` if the spaces of differentials $D1$ and $D2$ are the same. ## Operations on Elements A number of general operations for elements are also provided for differentials as well as a number of specific functions for differentials. ### Arithmetic Operators #### `r * x: RngElt, DiffFunElt -> DiffFunElt` #### `x * r: DiffFunElt, RngElt -> DiffFunElt` #### `x + y: DiffFunElt, DiffFunElt -> DiffFunElt` #### `- x: DiffFunElt -> DiffFunElt` #### `x - y: DiffFunElt, DiffFunElt -> DiffFunElt` #### `x / y: DiffFunElt, DiffFunElt -> FldFunGElt` #### `x / r: DiffFunElt, RngElt -> DiffFunElt` 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 $r \in F$ such that $x = ry$. ### Equality and Membership #### `x eq y: DiffFunElt, DiffFunElt -> BoolElt` Returns `true` if $x$ and $y$ are the same differential. #### `x in D: Any, DiffFun -> BoolElt` Returns `true` if $x$ is in the space of differentials $D$. ### Predicates on Elements #### `IsExact(d): DiffFunElt -> BoolElt, FldFunGElt` 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. #### `IsZero(d): DiffFunElt -> BoolElt` Return `true` if $d$ is the zero differential. ### Functions on Elements #### `Valuation(d, P): DiffFunElt, PlcFunElt -> RngIntElt` The valuation of the differential $d$ at the place $P$. #### `Divisor(d): DiffFunElt -> DivFunElt` The divisor $(d)$ of the differential $d$. #### `Residue(d, P): DiffFunElt, PlcFunElt -> RngElt` The residue of the differential $d$ at the place $P$, where $P$ must (currently) have degree one. #### `Example: Diff Fun (ex-7c77de)` ```magma > PF := PolynomialRing(GF(31, 3)); > P := PolynomialRing(PF); > FF1 := ext; > P := PolynomialRing(FF1); > FF2 := ext; > 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 ``` #### `Module(L, R): SeqEnum[ DiffFunElt ], Rng -> Mod, Map, SeqEnum[ ModElt ]` ```magma 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. #### `Relations(L, R): SeqEnum[ DiffFunElt ], Rng -> ModTupRng` #### `Relations(L, R, m): SeqEnum[ DiffFunElt ], Rng, RngIntElt -> ModTupRng` 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 $a_1, \dots, a_n$, $V$ be the relation module $\subseteq R^n$ and define $M := \{\, \sum_{i=1}^m v_i a_i \,|\, v = (v_i)_i \in V \,\}$. The function tries to compute a generating system of $V$ such that the corresponding generating system of $M$ consists of “small” elements. #### `Example: Module Diff (ex-3e031a)` This example shows some of the conversions and operations possible with the results of `Module`. ```magma > Q := Rationals(); > Qx := PolynomialRing(Q); > Qxy := 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) ``` #### `Cartier(b): DiffFunElt -> DiffFunElt` #### `Cartier(b, r): DiffFunElt, RngIntElt -> DiffFunElt` The result of applying the Cartier operator $r$ times to $b$. More precisely, let $F/k$ be a function field over the perfect field $k$, $x \in F$ be a separating variable and $b = g\,dx\,\in \Omega(F/k)$ with $g \in F$ be a differential. The Cartier operator is defined by $C(b) = \left( - d^{p-1} g / dx^{p-1} \right)^{1/p} dx.$ This function computes the $r$-th iterated application of $C$ to $b$. ### Other #### `CartierRepresentation(F): FldFunG -> AlgMatElt, SeqEnum[DiffFunElt]` #### `CartierRepresentation(F, r): FldFunG, RngIntElt -> AlgMatElt, SeqEnum[DiffFunElt]` 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 $k$, $\omega_1, \dots, \omega_g \in \Omega(F/k)$ be a basis for the holomorphic differentials and $r \in {\mathbb{Z}}^{\geq 1}$. Let $M = (\lambda_{i,j})_{i,j} \in k^{g \times g}$ be the matrix such that $$ C^r(\omega_i) = \sum_{m=1}^g \lambda_{i, m} \omega_m $$ for all $1 \leq i \leq g$. This function returns $M$ and $(\omega_1, \dots, \omega_g)$. #### `Example: Diff Cart (ex-78c02f)` An example of a trivial cartier action. ```magma > PF := PolynomialRing(GF(31, 3)); > P := PolynomialRing(PF); > FF1 := ext; > P := PolynomialRing(FF1); > FF2 := ext; > Cartier(Differential(d), 4); (0) d(x) > CartierRepresentation(FF2, 3); Matrix with 0 rows and 0 columns [] ```