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Provide links for some documentation references #3503

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4 changes: 2 additions & 2 deletions docs/src/AlgebraicGeometry/ToricVarieties/AlgebraicCycles.md
Original file line number Diff line number Diff line change
Expand Up @@ -61,7 +61,7 @@ rational_equivalence_class(v::NormalToricVarietyType, p::MPolyQuoRingElem)
rational_equivalence_class(v::NormalToricVarietyType, coefficients::Vector{T}) where {T <: IntegerUnion}
```

### Special constructors
### [Special constructors](@id toric_special_constructors)

```@docs
rational_equivalence_class(d::ToricDivisor)
Expand Down Expand Up @@ -95,7 +95,7 @@ An algebraic cycle can be intersected `n`- with itself via `^n`,
where `n` can be an integer of of type `ZZRingElem`.

A closed subvarieties defines in a natural way a rational equivalence
class (cf. section on special constructors above). This allows to
class (cf. [Special constructors](@ref toric_special_constructors)). This allows to
compute intersection products among closed subvarieties and rational
equivalence classes in the Chow ring.

Expand Down
6 changes: 3 additions & 3 deletions docs/src/CommutativeAlgebra/GroebnerBases/groebner_bases.md
Original file line number Diff line number Diff line change
Expand Up @@ -5,7 +5,7 @@ DocTestSetup = quote
end
```

# Gröbner/Standard Bases Over Fields
# [Gröbner/Standard Bases Over Fields](@id gb_fields)

We fix our notation in the context of standard (Gröbner) bases and present relevant OSCAR functions.

Expand Down Expand Up @@ -77,7 +77,7 @@ julia> default_ordering(S)
wdegrevlex([x, y, z], [1, 2, 3])
```

## Monomials, Terms, and More
## [Monomials, Terms, and More](@id monomials_terms_more)

Here are examples which indicate how to recover monomials, terms, and
more from a given polynomial.
Expand Down Expand Up @@ -400,7 +400,7 @@ normal_form(g::T, I::MPolyIdeal; ordering::MonomialOrdering = default_ordering(b

## Syzygies

We refer to the section on modules for more on syzygies.
We refer to the section on [modules](@ref modules_multivariate) for more on syzygies.

```@docs
syzygy_generators(G::Vector{<:MPolyRingElem})
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Expand Up @@ -33,7 +33,7 @@ julia> reduce(6*x, [5*x, 2*x])
0
```

The notion of *leading ideals* as formulated in the previous section and the definitions of
The notion of *leading ideals* as formulated in [the previous section](@ref gb_fields) and the definitions of
standard bases (Gröbner bases) carry over: A *standard basis* for an ideal $I\subset K[x]_>$
with respect to $>$ is a finite subset $G$ of $I$ such that $\text{L}_>(G) = \text{L}_>(I)$ (a
standard basis with respect to a global monomial ordering is also called a *Gröbner basis*).
Expand Down
4 changes: 2 additions & 2 deletions docs/src/CommutativeAlgebra/GroebnerBases/orderings.md
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Expand Up @@ -5,7 +5,7 @@ DocTestSetup = quote
end
```

# Monomial Orderings
# [Monomial Orderings](@id monomial_orderings)

Given a coefficient ring $C$ as in the previous section, let $C[x]=C[x_1, \ldots, x_n]$
be the polynomial ring over $C$ in the set of variables $x=\{x_1, \ldots, x_n\}$. Monomials
Expand Down Expand Up @@ -37,7 +37,7 @@ A monomial ordering $>$ on $\text{Mon}_n(x)$ is called
!!! note
The lexicograpical monomial ordering specifies the default way of storing and displaying multivariate polynomials in OSCAR (terms are sorted in descending order).
The other orderings which can be attached to a multivariate polynomial ring are the degree lexicographical ordering and the degree reverse lexicographical
ordering. Independently of the attached orderings, Gröbner bases can be computed with respect to any monomial ordering. See the section on Gröbner bases.
ordering. Independently of the attached orderings, Gröbner bases can be computed with respect to any monomial ordering. See the section on [Gröbner bases](@ref gb_fields).

In this section, we show how to create monomial orderings in OSCAR.

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Expand Up @@ -31,7 +31,7 @@ While the first step can be performed over any ground field for which Gröbner b
!!! note
A *pure difference binomial* is a binomial which is the difference of two monomials. A *unital binomial ideal* is an ideal which can be generated by pure difference binomials and monomials. Note that cellular components of unital binomial ideals are unital as well. For unital binomial ideals in $\mathbb Q[x_1, \dots, x_n]$, binomial primary decompositions exist already over cyclotomic extensions of $\mathbb Q$. In particular, any such ideal can be decomposed over the abelian closure $\mathbb Q^{\text{ab}}$ of $\mathbb Q$. While OSCAR offers functionality for doing this, computing binomial primary decompositions in other cases is not yet supported.

See the number theory chapter for how to deal with $\mathbb Q^{\text{ab}}$.
See the [number theory](@ref number_theory) chapter for how to deal with $\mathbb Q^{\text{ab}}$.
!!! note
Binomial primary decompositions computed with OSCAR are not necessarily minimal.

Expand Down
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Expand Up @@ -5,7 +5,7 @@ DocTestSetup = quote
end
```

# Free Modules
# [Free Modules](@id free_modules)

In this section, the expression *free module* refers to a free module of finite rank
over a ring of type `MPolyRing`, `MPolyQuoRing`, `MPolyLocRing`, or `MPolyQuoLocRing`.
Expand Down
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Expand Up @@ -2,7 +2,7 @@
CurrentModule = Oscar
```

# Introduction
# [Introduction](@id modules_multivariate)

Our focus in this section is on finitely presented modules over rings from the following list:
- multivariate polynomial rings (OSCAR type `MPolyRing`),
Expand All @@ -17,8 +17,8 @@ modules which naturally includes both submodules and quotients of free modules.

!!! note
Most functions in this section rely on Gröbner (standard) bases techniques. Thus, the functions
should not be applied to modules over rings other than those from the list above. See the Linear
Algebra chapter for module types which are designed to handle modules over Euclidean
should not be applied to modules over rings other than those from the list above. See the [Linear
Algebra](@ref linear_algebra) chapter for module types which are designed to handle modules over Euclidean
domains.

!!! note
Expand Down
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Expand Up @@ -34,7 +34,7 @@ Finally, we refer to
and regard $M$ as a submodule of that ambient module, embedded in the natural way.

!!! note
Recall from the section on free modules that by a free $R$-module we mean a free
Recall from the section on [free modules](@ref free_modules) that by a free $R$-module we mean a free
module of type $R^p$ , where we think of $R^p$ as a free module with a given
basis, namely the basis of standard unit vectors. Accordingly, elements of free modules
are represented by coordinate vectors, and homomorphisms between free modules by
Expand Down
8 changes: 4 additions & 4 deletions docs/src/CommutativeAlgebra/affine_algebras.md
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Expand Up @@ -5,7 +5,7 @@ DocTestSetup = quote
end
```

# Affine Algebras and Their Ideals
# [Affine Algebras and Their Ideals](@id affine_algebras)

With regard to notation, we use *affine algebra* as a synonym for *quotient of a multivariate polynomial ring by an ideal*.
More specifically, if $R$ is a multivariate polynomial ring with coefficient ring $C$, and $A=R/I$ is the quotient of $R$
Expand All @@ -14,14 +14,14 @@ discuss functionality for handling such algebras in OSCAR.

!!! note
To emphasize this point: In this section, we view $R/I$ together with its ring structure. Realizing $R/I$ as an
$R$-module means to implement it as the quotient of a free $R$-module of rank 1. See the section on modules.
$R$-module means to implement it as the quotient of a free $R$-module of rank 1. See the section on [modules](@ref modules_multivariate).

!!! note
Most functions discussed here rely on Gröbner basis techniques. In particular, they typically make use of a Gröbner basis for the
modulus of the quotient. Nevertheless, the construction of quotients is lazy in the sense that the computation of such a Gröbner
basis is delayed until the user performs an operation that indeed requires it (the Gröbner basis is then computed with respect
to the monomial ordering entered by the user when creating the quotient; if no such ordering is entered, OSCAR will use the
`default_ordering` on the underlying polynomial ring; see the section on Gröbner/Standard Bases for default orderings in OSCAR).
`default_ordering` on the underlying polynomial ring; see the section on [Gröbner/Standard Bases](@ref gb_fields) for default orderings in OSCAR).
Once computed, the Gröbner basis is cached for later reuse.

!!! note
Expand All @@ -31,7 +31,7 @@ discuss functionality for handling such algebras in OSCAR.
!!! note
In OSCAR, elements of a quotient $A = R/I$ are not necessarily represented by polynomials which are reduced with regard to $I$.
That is, if $f\in R$ is the internal polynomial representative of an element of $A$, then $f$ may not be the normal form mod $I$
with respect to the default ordering on $R$ (see the section on *Gröbner/Standard Bases* for normal forms). Operations involving
with respect to the default ordering on $R$ (see the section on [Gröbner/Standard Bases](@ref gb_fields) for normal forms). Operations involving
Gröbner basis computations may lead to (partial) reductions. The function `simplify` discussed in this section computes fully
reduced representatives.

Expand Down
4 changes: 2 additions & 2 deletions docs/src/CommutativeAlgebra/intro.md
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Expand Up @@ -2,7 +2,7 @@
CurrentModule = Oscar
```

# Introduction
# [Introduction](@id commutative_algebra)

The commutative algebra part of OSCAR provides functionality for dealing with

Expand All @@ -15,7 +15,7 @@ We use *affine algebra* as a synonym for *quotient of a multivariate polynomial

Fundamental to computational commutative algebra is the concept of *standard bases*. Each such basis
is defined relative to a *monomial ordering*. If this ordering is a well-ordering, a standard basis is also called
a *Gröbner basis*. We refer to the corresponding section in this chapter for details.
a *Gröbner basis*. We refer to the corresponding [section](@ref gb_fields) in this chapter for details.

!!! note
Each multivariate polynomial ring in OSCAR comes equipped with a monomial ordering according to which the
Expand Down
2 changes: 1 addition & 1 deletion docs/src/CommutativeAlgebra/localizations.md
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Expand Up @@ -23,7 +23,7 @@ Mimicking the standard arithmetic for fractions, ``R[U^{-1}]`` can be made into
\iota : R\to R[U^{-1}],\; r \mapsto \frac{r}{1}.
```
Given an ``R``-module ``M``, the analogous construction yields an ``R[U^{-1}]``-module ``M[U^{-1}]`` which is
called the *localization of ``M`` at ``U``*. See the section on modules.
called the *localization of ``M`` at ``U``*. See the section on [modules](@ref modules_multivariate).

Our focus in this section is on localizing multivariate polynomial rings and their quotients. The starting point
for this is to provide functionality for handling (several types of) multiplicatively closed subsets of multivariate
Expand Down
9 changes: 5 additions & 4 deletions docs/src/CommutativeAlgebra/rings.md
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Expand Up @@ -7,8 +7,9 @@ end

# Creating Multivariate Rings

In this section, for the convenience of the reader, we recall from the chapters on rings and fields
how to create multivariate polynomial rings and their elements, adding illustrating examples.
In this section, for the convenience of the reader, we recall from the chapters on
[rings](@ref rings) and [fields](@ref fields) how to create multivariate polynomial
rings and their elements, adding illustrating examples.
At the same time, we introduce and illustrate a ring type for modelling multivariate polynomial
rings with gradings.

Expand Down Expand Up @@ -432,7 +433,7 @@ Given an element `f` of a multivariate polynomial ring `R` or a graded version o
the notion of total degree ignores the weights given to the variables in the graded case.

For iterators which allow one to recover the monomials (terms, $\dots$) of `f` we refer to the
subsection *Monomials, Terms, and More* of the section on *Gröbner/Standard Bases*.
subsection [Monomials, Terms, and More](@ref monomials_terms_more) of the section on [Gröbner/Standard Bases](@ref gb_fields).

###### Examples

Expand Down Expand Up @@ -502,5 +503,5 @@ refer to `R` and `S`, respectively.
!!! note
The OSCAR homomorphism type `AffAlgHom` models ring homomorphisms `R` $\to$ `S` such that
the type of both `R` and `S` is a subtype of `Union{MPolyRing{T}, MPolyQuoRing{U}}`, where `T <: FieldElem` and
`U <: MPolyRingElem{T}`. Functionality for these homomorphism is discussed in the section on affine algebras.
`U <: MPolyRingElem{T}`. Functionality for these homomorphism is discussed in the section on [affine algebras](@ref affine_algebras).

4 changes: 2 additions & 2 deletions docs/src/DeveloperDocumentation/new_developers.md
Original file line number Diff line number Diff line change
Expand Up @@ -75,7 +75,7 @@ in Julia. This will create a directory `~/.julia/dev/Oscar`. This directory is
a git clone of the central OSCAR repository. You can develop your code here,
however you will still have to fork OSCAR, as you have no rights to push to the
central repository. You can then add your fork as another remote, have a look
at the section on rebasing below for hints.
at the section on [rebasing](@ref rebasing) below for hints.


## The edit process
Expand Down Expand Up @@ -160,7 +160,7 @@ naming conventions, code formatting, etc.
To build and test the documentation, please have a look at [Documenting OSCAR
code](@ref).

### Rebasing
### [Rebasing](@id rebasing)
One way to stay up to date with the current master is rebasing. In order to do
this, add the main Oscar.jl repository as a remote, fetch, and then rebase.
```
Expand Down
2 changes: 1 addition & 1 deletion docs/src/Fields/intro.md
Original file line number Diff line number Diff line change
Expand Up @@ -5,7 +5,7 @@ DocTestSetup = quote
end
```

# Introduction
# [Introduction](@id fields)

The fields part of OSCAR provides functionality for handling
various kinds of fields:
Expand Down
2 changes: 1 addition & 1 deletion docs/src/InvariantTheory/finite_groups.md
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Expand Up @@ -5,7 +5,7 @@ DocTestSetup = Oscar.doctestsetup()

# Invariants of Finite Groups

In this section, with notation as in the introduction to this chapter, $G$ will be a *finite* group.
In this section, with notation as in the [introduction](@ref invariant_theory) to this chapter, $G$ will be a *finite* group.

!!! note
The ssumption that $G$ is finite implies:
Expand Down
2 changes: 1 addition & 1 deletion docs/src/InvariantTheory/intro.md
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Expand Up @@ -2,7 +2,7 @@
CurrentModule = Oscar
```

# Introduction
# [Introduction](@id invariant_theory)

The invariant theory part of OSCAR provides functionality for computing polynomial invariants
of group actions, focusing on finite and linearly reductive groups, respectively.
Expand Down
2 changes: 1 addition & 1 deletion docs/src/InvariantTheory/reductive_groups.md
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Expand Up @@ -4,7 +4,7 @@ CurrentModule = Oscar

# Invariants of Linearly Reductive Groups

In this section, with notation as in the introduction to this chapter,
In this section, with notation as in the [introduction](@ref invariant_theory) to this chapter,
$G$ will be a *linearly algebraic group* over an algebraically closed
field $K$, $\rho: G \to \text{GL}(V)\cong \text{GL}_n(K)$ will
be a *rational* representation of $G$, and
Expand Down
2 changes: 1 addition & 1 deletion docs/src/InvariantTheory/tori.md
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Expand Up @@ -3,7 +3,7 @@ CurrentModule = Oscar
```

# Invariants of Tori
In this section, with notation as in the introduction to this chapter, $T =(K^{\ast})^m$ will be a torus of rank $m$
In this section, with notation as in the [introduction](@ref invariant_theory) to this chapter, $T =(K^{\ast})^m$ will be a torus of rank $m$
over a field $K$. To compute invariants of diagonal torus actions, OSCAR makes use of Algorithm 4.3.1 in [DK15](@cite) which,
in particular, relies on algorithmic means from polyhedral geometry.

Expand Down
2 changes: 1 addition & 1 deletion docs/src/LinearAlgebra/intro.md
Original file line number Diff line number Diff line change
Expand Up @@ -5,7 +5,7 @@ DocTestSetup = quote
end
```

# Introduction
# [Introduction](@id linear_algebra)

The linear algebra part of OSCAR provides functionality for handling
- vectors and matrices
Expand Down
2 changes: 1 addition & 1 deletion docs/src/NoncommutativeAlgebra/PBWAlgebras/intro.md
Original file line number Diff line number Diff line change
Expand Up @@ -29,7 +29,7 @@ x_jx_i = c_{ij}x_ix_j+d_{ij}.
```

Working with Gröbner bases requires that we take monomial orderings into account (see the section
on Gröbner bases in the commutative algebra chapter for monomial orderings). In our context here, we use the following notation.
on [Gröbner bases](@ref monomial_orderings) in the [commutative algebra](@ref commutative_algebra) chapter for monomial orderings). In our context here, we use the following notation.
A *standard monomial* in $K \langle x \rangle$ is a word of type $x^\alpha=x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}},$
where $\alpha=(\alpha_1,\dots,\alpha_n)\in\mathbb N^n$. A *standard polynomial* in $K \langle x \rangle$
is a $K$-linear combination of standard monomials. Each global monomial ordering $>$ on $K[x]$ gives rise to
Expand Down
2 changes: 1 addition & 1 deletion docs/src/NoncommutativeAlgebra/PBWAlgebras/quotients.md
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Expand Up @@ -7,7 +7,7 @@ end

# GR-Algebras: Quotients of PBW-Algebras

In analogy to the affine algebras section in the commutative algebra chapter, we describe OSCAR
In analogy to the [affine algebras](@ref affine_algebras) section in the [commutative algebra](@ref commutative_algebra) chapter, we describe OSCAR
functionality for dealing with quotients of PBW-algebras modulo two-sided ideals.

!!! note
Expand Down
2 changes: 1 addition & 1 deletion docs/src/NumberTheory/intro.md
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Expand Up @@ -2,7 +2,7 @@
CurrentModule = Oscar
```

# Introduction
# [Introduction](@id number_theory)

The number theory part of OSCAR provides functionality for algebraic number theory.

Expand Down
2 changes: 1 addition & 1 deletion docs/src/Rings/intro.md
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Expand Up @@ -5,7 +5,7 @@ DocTestSetup = quote
end
```

# Introduction
# [Introduction](@id rings)

The rings part of OSCAR provides functionality for handling
various kinds of rings:
Expand Down
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