# General Structure Invariants ## `Characteristic(F): FldFun -> RngIntElt` ## `Characteristic(O): RngFunOrd -> RngIntElt` The characteristic of the function field $F/k$ or one of its orders $O$. ## `IsPerfect(F): Fld -> BoolElt` Applies to any field in Magma. Returns whether $F$ is perfect. ## `Degree(F): FldFunG -> RngIntElt` ## `Degree(F, G): FldFun, FldFun -> RngIntElt` ## `Degree(O): RngFunOrd -> RngIntElt` The degree $[F:G]$ of the field extension $F/G$ where $G$ is the base field of $F$ unless specified. For an order $O$, this function returns the rank of $O$ as a module over its coefficient ring. Note that this rank is equal to the degree $[F:G]$ where $F$ and $G$ are the field of fractions of $O$ and the coefficient ring of $O$ respectively. ## `AbsoluteDegree(F): FldFunG -> RngIntElt` ## `AbsoluteDegree(O): RngFunOrd -> RngIntElt` The degree of the function field $F$ or the order $O$ as a finite extension of $k(x)$ or $k[x]$ or as an infinite extension of $k$. ## `DefiningPolynomial(F): FldFun -> RngUPolElt` ## `DefiningPolynomial(O): RngFunOrd -> RngUPolElt` The defining polynomial of the function field $F$ over its coefficient ring. For an order $O$ belonging to a function field $F$, this function returns the defining polynomial of $O$, which may be different from that of $F/k(x, \alpha_1, \ldots, \alpha_r)$. ## `DefiningPolynomials(F): FldFun -> [RngUPolElt]` ## `DefiningPolynomials(O): RngFunOrd -> [RngUPolElt]` Return the defining polynomials of the function field $F$ or the order $O$ as a sequence of polynomials over the coefficient ring. ## `Basis(F): FldFunG -> SeqEnum[FldFunElt]` ## `Basis(O): RngFunOrd -> SeqEnum[FldFunElt]` ## `Basis(O, R): RngFunOrd, Rng -> SeqEnum[RngElt]` The basis $1, \alpha, \dots, \alpha^{n-1}$ of the function field $F[\alpha]$ over the coefficient field. Given an order $O$ belonging to a function field $F$, this function returns the basis of $O$ in the form of function field elements. Given an additional ring $R$, return the basis of $O$ as elements of $R$. ## `TransformationMatrix(O1, O2): RngFunOrd, RngFunOrd -> AlgMatElt, RngElt` Return the matrix $M$ and a denominator $d$ which transforms elements of the order $O1$ into elements of the order $O2$. ## `CoefficientIdeals(O): RngFunOrd -> [RngFunOrdIdl]` The coefficient ideals of the order $O$ of a relative extension. These are the ideals $\{A_i\}$ of the coefficient ring of $O$ such that for every element $e$ of $O$, $e = \sum_i a_i*b_i$ where $\{b_i\}$ is the basis returned for $O$ and each $a_i \in A_i$. ## `BasisMatrix(O): RngFunOrd -> AlgMatElt` Given an order $O$ in a function field $F$ of degree $n$, this returns an $n\times n$ matrix whose $i$-th row contains the coefficients for the $i$-th basis element of $O$ with respect to the power basis of $F$. Thus, if $b_i$ is the $i$-th basis element of $O$, $$ b_i=\sum_{j=1}^{n}M_{ij}\alpha^{j-1} $$ where $M$ is the matrix and $\alpha$ is the generator of $F$. ## `PrimitiveElement(O): RngFunOrd -> RngFunOrdElt` A root of the defining polynomial of the order $O$. ## `Discriminant(O): RngFunOrd -> .` The discriminant of the order $O$, up to a unit in its coefficient ring. ## `AbsoluteDiscriminant(O): RngFunOrd -> .` The discriminant of the order $O$ of an algebraic function field $F$ over the bottom coefficient ring of $O$, (the subring of the rational function field $F$ extends). ## `DimensionOfExactConstantField(F): FldFunG -> RngIntElt` ## `DegreeOfExactConstantField(F): FldFunG -> RngIntElt` The dimension of the exact constant field of the function field $F/k$ over $k$. The exact constant field is the algebraic closure of $k$ in $F$. ## `Genus(F): FldFunG -> RngIntElt` ```magma Al : MonStgElt Default: IsExact: BoolElt Default: false ``` The genus of the function field $F/k$. If $F$ is an extension of a rational function field over ${\mathbb{Q}}$ or ${\mathbb{F}}_q$ by a single monic integral polynomial and the parameter `Al` is set to `"Montes"` then the Montes algorithm [[Stainsby, 2018](../../references.md#cite-2015arxiv150601904s)] will be used to compute the genus. The index $[k_0:{\rm ConstantField}(F)]$, where $k_0$ is the full constant field, is also returned. If it is known beforehand that $k_0$ is the constant field $F$ is defined over then set the parameter `IsExact := true`. ## `Example: invar (ex-a8d7ef)` ```magma > PF := PolynomialRing(GF(31, 3)); > P := PolynomialRing(PF); > FF1 := ext; > P := PolynomialRing(FF1); > FF2 := ext; > Characteristic(FF2); 31 > EFF2I := EquationOrderInfinite(FF2); > MFF2I := MaximalOrderInfinite(FF2); > Degree(MFF2I) eq 3; true > AbsoluteDegree(EFF2I); 6 > Genus(FF2); 9 > DefiningPolynomial(EFF2I); $.1^3 + [ 0, 30/x^3 ]*$.1 + [ 30/x^9, 0 ] > Basis(MFF2I); [ 1, 1/x*d, 1/x^2*d^2 ] > Discriminant(EFF2I); Ideal of Maximal Equation Order of FF1 over Valuation ring of Univariate rational function field over GF(31^3) Variables: x with generator 1/x Generator: (4*x^3 + 27)/x^15*b + 4/x^18 > AbsoluteOrder(EFF2I); Order of Algebraic function field defined over Univariate rational function field over GF(31^3) by y^6 + 29*y^3 + (30*x^5 + x^2)*y^2 + 1 over Valuation ring of Univariate rational function field over GF(31^3) with generator 1/x > AbsoluteDiscriminant(EFF2I); (2*x^9 + 25*x^6 + 6*x^3 + 29)/x^33 > Discriminant($2); (30*x^24 + 6*x^21 + 16*x^18 + 20*x^15 + 16*x^12 + 7*x^9 + 27*x^6 + 3*x^3 + 30)/x^48 ``` ## `Example: Invar Non Simple (ex-ec93f9)` Invariants are slightly different for non–simple fields. ```magma > P := PolynomialRing(Rationals()); > P := PolynomialRing(P); > F := FunctionField([3*y^3 - x^2, x*y^2 + 1]); > DefiningPolynomials(F); [ 3*y^3 - x^2, x*y^2 + 1 ] > DefiningPolynomials(EquationOrderFinite(F)); [ y^3 - 1/3*x^2, y^2 + x ] > DefiningPolynomials(EquationOrderInfinite(F)); [ $.1^3 - 1/3/$.1^4, $.1^2 + 1/$.1 ] > Basis(F); [ 1, a, a^2, $.1*b, $.1*a*b, $.1*a^2*b ] > TransformationMatrix(EquationOrderFinite(F), MaximalOrderFinite(F)); [1 0 0 0 0 0] [0 1 0 0 0 0] [0 0 x 0 0 0] [0 0 0 1 0 0] [0 0 0 0 x 0] [0 0 0 0 0 x] 1 > TransformationMatrix(MaximalOrderFinite(F), EquationOrderFinite(F)); [x 0 0 0 0 0] [0 x 0 0 0 0] [0 0 1 0 0 0] [0 0 0 x 0 0] [0 0 0 0 1 0] [0 0 0 0 0 1] x ``` ## `GapNumbers(F): FldFunG -> SeqEnum[RngIntElt]` ```magma SeparatingElement: FldFunGElt Default: ``` The sequence of global gap numbers of the function field $F/k$ (in characteristic zero this is always $[1, \dots, g]$). A separating element used internally for the computation can be specified, it defaults to `SeparatingElement(F)`. See the description of [`GapNumbers`](divisors.md#function-divfunelt-gapnumbers). ## `GapNumbers(F, P): FldFunG, PlcFunElt -> SeqEnum[RngIntElt]` ## `GapNumbers(P): PlcFunElt -> SeqEnum[RngIntElt]` The sequence of gap numbers of the function field $F/k$ at $P$ where $P$ must be a place of degree one. See the description of [`GapNumbers`](divisors.md#function-divfunelt-gapnumbers). ## `SeparatingElement(F): FldFunG -> FldFunGElt` Returns a separating element of the function field $F/k$. ## `RamificationDivisor(F): FldFunG -> DivFunElt` ```magma SeparatingElement: FldFunGElt Default: ``` The ramification divisor of the function field $F/k$. The semantics of calling `RamificationDivisor()` with $F$ or the zero divisor of $F$ are identical. For further details see the description of [`RamificationDivisor`](divisors.md#function-divfunelt-ramificationdivisor). ## `WeierstrassPlaces(F): FldFunG -> [PlcFunElt]` ```magma SeparatingElement: FldFunGElt Default: ``` The Weierstrass places of the function field $F/k$. The semantics of calling `WeierstrassPlaces` with $F$ or the zero divisor of $F$ are identical. See the description of [`WeierstrassPlaces`](divisors.md#function-divfunelt-weierstrassplaces). ## `WronskianOrders(F): FldFunG -> [RngIntElt]` ```magma SeparatingElement: FldFunGElt Default: ``` The Wronskian orders of the function field $F/k$. The semantics of calling `WronskianOrders` with $F$ or the zero divisor of $F$ are identical. See the description of [`WronskianOrders`](divisors.md#function-divfunelt-wronskianorders). ## `Different(O): RngFunOrd -> RngFunOrdIdl` The different of the maximal order $O$. ## `Index(O, S): RngFunOrd, RngFunOrd -> Any` The index of $S$ in $O$ where $S$ is a suborder of $O$ and $O$ and $S$ have the same equation order.