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#

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] 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)#

> PF<x> := PolynomialRing(GF(31, 3));
> P<y> := PolynomialRing(PF);
> FF1<b> := ext<FieldOfFractions(PF) | y^2 - x^3 + 1>;
> P<y> := PolynomialRing(FF1);
> FF2<d> := ext<FF1 | y^3 - b*x*y - 1>;
> 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

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Example: Invar Non Simple (ex-ec93f9)#

Invariants are slightly different for non–simple fields.

> P<x> := PolynomialRing(Rationals());
> P<y> := PolynomialRing(P);
> F<a, b> := 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

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GapNumbers(F): FldFunG -> SeqEnum[RngIntElt]#

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.

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.

SeparatingElement(F): FldFunG -> FldFunGElt#: Returns a separating element of the function field \(F/k\).

RamificationDivisor(F): FldFunG -> DivFunElt#

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.

WeierstrassPlaces(F): FldFunG -> [PlcFunElt]#

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.

WronskianOrders(F): FldFunG -> [RngIntElt]#

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.

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.