More references, tests, small things

docs/oscar_references.bib

@@ -47,3 +47,87 @@ @book{DP13
 year={2013},
 collection={AIMS Library of Mathematical Sciences}
 }
+
+@InProceedings{KL91,
+author="Krick, Teresa
+and Logar, Alessandro",
+editor="Mattson, Harold F.
+and Mora, Teo
+and Rao, T. R. N.",
+title="An algorithm for the computation of the radical of an ideal in the ring of polynomials",
+booktitle="Applied Algebra, Algebraic Algorithms and Error-Correcting Codes",
+year="1991",
+publisher="Springer Berlin Heidelberg",
+address="Berlin, Heidelberg",
+pages="195--205",
+isbn="978-3-540-38436-6"
+}
+
+@article{Kem02,
+title = {The Calculation of Radical Ideals in Positive Characteristic},
+journal = {Journal of Symbolic Computation},
+volume = {34},
+number = {3},
+pages = {229-238},
+year = {2002},
+issn = {0747-7171},
+doi = {https://doi.org/10.1006/jsco.2002.0560},
+url = {https://www.sciencedirect.com/science/article/pii/S0747717102905602},
+author = {Gregor Kemper},
+abstract = {We propose an algorithm for computing the radical of a polynomial ideal in positive characteristic. The algorithm does not involve polynomial factorization.}
+}
+
+@article{PSS11,
+author = {Gerhard Pfister and Afshan Sadiq and Stefan Steidel},
+doi = {doi:10.2478/s11533-011-0037-8},
+url = {https://doi.org/10.2478/s11533-011-0037-8},
+title = {An algorithm for primary decomposition in polynomial rings over the integers},
+journal = {Open Mathematics},
+number = {4},
+volume = {9},
+year = {2011},
+pages = {897--904}
+}
+
+@article{GTZ88,
+title = {Gröbner bases and primary decomposition of polynomial ideals},
+journal = {Journal of Symbolic Computation},
+volume = {6},
+number = {2},
+pages = {149-167},
+year = {1988},
+issn = {0747-7171},
+doi = {https://doi.org/10.1016/S0747-7171(88)80040-3},
+url = {https://www.sciencedirect.com/science/article/pii/S0747717188800403},
+author = {Patrizia Gianni and Barry Trager and Gail Zacharias},
+abstract = {We present an algorithm to compute the primary decomposition of any ideal in a polynomialring over a factorially closed algorithmic principal ideal domain R. This means that the ring R is a constructive PID and that we are given an algorithm to factor polynomials over fields which are finitely generated over R or residue fields of R. We show how basic ideal theoretic operations can be performed using Gröbner bases and we exploit these constructions to inductively reduce the problem to zero dimensional ideals. Here we again exploit the structure of Gröbner bases to directly compute the primary decomposition using polynomial factorization. We also show how the reduction process can be applied to computing radicals and testing ideals for primality.}
+}
+
+@article{SY96,
+title = {Localization and Primary Decomposition of Polynomial Ideals},
+journal = {Journal of Symbolic Computation},
+volume = {22},
+number = {3},
+pages = {247-277},
+year = {1996},
+issn = {0747-7171},
+doi = {https://doi.org/10.1006/jsco.1996.0052},
+url = {https://www.sciencedirect.com/science/article/pii/S0747717196900528},
+author = {Takeshi Shimoyama and Kazuhiro Yokoyama},
+abstract = {In this paper, we propose a new method for primary decomposition of a polynomial ideal, not necessarily zero-dimensional, and report on a detailed study for its practical implementation. In our method, we introduce two key techniques,effective localizationandfast elimination of redundant components, by which we can get a good performance for several examples. The performance of our method is examined by comparison with other existing methods based on practical experiments.}
+}
+
+@article{EHV92,
+ author = {David Eisenbud and Craig Huneke and Wolmer Vasconcelos},
+ title = {Direct methods for primary decomposition},
+ journal = {Inventiones Mathematicae},
+ issn = {0020-9910; 1432-1297/e},
+ volume = {110},
+ number = {2},
+ pages = {207--235},
+ year = {1992},
+ publisher = {Springer, Berlin/Heidelberg},
+ language = {English},
+ MSC2010 = {13P10 13A05 13A15 13F20 13A10},
+ Zbl = {0770.13018}
+}

docs/src/CommutativeAlgebra/ca_affine_algebras.md

@@ -20,24 +20,16 @@ polynomial ring $R$ over $K$ modulo an ideal $I$ of $A$.
 ```@docs
 quo(R::MPolyRing, I::MPolyIdeal)
 ```
-###### Example
-
-```@repl oscar
-R, (x, y) = PolynomialRing(QQ, ["x", "y"])
-A, _ = quo(R, ideal(R, [x^2-y^3, x-y]))
-A, f = quo(R, ideal(R, [x^2-y^3, x-y]))
-f
-```
 
 ## Data Associated to Affine Algebras
 
 ### Basic Data
 
 If `A = R/I` is an affine algebra, then
-- base_ring(A) refers to R,
-- modulus(A) to I,
-- gens(A) to the generators of A, and
-- ngens(A) to the number of these generators.
+- `base_ring(A)` refers to `R`,
+- `modulus(A)` to `I`,
+- `gens(A)` to the generators of `A`, and
+- `ngens(A)` to the number of these generators.
 
 ```@repl oscar
 R, (x, y, z) = PolynomialRing(QQ, ["x", "y", "z"])
@@ -54,14 +46,6 @@ ngens(A)
 dim(A::MPolyQuo)
 ```
 
-###### Example
-
-```@repl oscar
-R, (x, y, z) = PolynomialRing(QQ, ["x", "y", "z"])
-A, _ = quo(R, ideal(R, [y-x^2, x-z^3]))
-dim(A)
-```
-
 ## Hilbert Series and Hilbert Polynomial
 
 Let $R=K[x_1, \dots x_n]$ be a polynomial ring in $n$ variables over a field $K$.

docs/src/CommutativeAlgebra/ca_ideals.md

@@ -23,7 +23,7 @@ ideal(Rx::MPolyRing, g::Array{<:Any, 1})
 
 ## Gröbner Bases
 
-### Monomial Orders
+### Monomial Orderings
 
 ### Normal Forms
 
@@ -102,16 +102,6 @@ dim(I::MPolyIdeal)
 codim(I::MPolyIdeal)
 ```
 
-## Hilbert Function and Hilbert Series
-
-                             min_base(I)   (or so)
-
-    degree:                  degree(I)                             (type integer)
-    Hilbert function:        hilbert_function(d,I)                 (type integer)
-    Hilbert series:          hilbert_series(I)                     (type univariate rational function)  numerator, denominator
-    reduced Hilbert series:  reduced_Hilbert_series(I)             (type univariate rational function)
-    Hilbert polynomial:      hilbert_polynomial(I)                 (type univariate polynomial, direkt in Julia von hilbert_series(I))
-
 ## Operations on Ideals
 
 ### Simple ideal Operations
@@ -162,11 +152,11 @@ saturation(I::MPolyIdeal, J::MPolyIdeal)
 ### Elimination
 
 ```@docs
-eliminate(I::MPolyIdeal, polys::Array{MPolyElem, 1})
+eliminate(I::MPolyIdeal, lv::Array{MPolyElem, 1})
 ```
 
 ```@docs
-eliminate(I::MPolyIdeal, polys::AbstractArray{Int, 1})
+eliminate(I::MPolyIdeal, li::AbstractArray{Int, 1})
 ```
 
 ### Homogenization and Dehomogenization

src/Rings/MPolyQuo.jl

@@ -258,6 +258,25 @@ end
 
 Creates the affine ring ``R/I`` and returns the new
 ring as well as the projection map $R\rightarrow R/I$.
+###### Example
+```jldoctest
+julia> R, (x, y) = PolynomialRing(QQ, ["x", "y"])
+(Multivariate Polynomial Ring in x, y over Rational Field, fmpq_mpoly[x, y])
+
+julia> A, _ = quo(R, ideal(R, [x^2-y^3, x-y]))
+(Quotient of Multivariate Polynomial Ring in x, y over Rational Field by ideal generated by: x^2 - y^3, x - y, Map from
+Multivariate Polynomial Ring in x, y over Rational Field to Quotient of Multivariate Polynomial Ring in x, y over Rational Field by ideal generated by: x^2 - y^3, x - y defined by a julia-function with inverse
+)
+
+julia> A, f = quo(R, ideal(R, [x^2-y^3, x-y]))
+(Quotient of Multivariate Polynomial Ring in x, y over Rational Field by ideal generated by: x^2 - y^3, x - y, Map from
+Multivariate Polynomial Ring in x, y over Rational Field to Quotient of Multivariate Polynomial Ring in x, y over Rational Field by ideal generated by: x^2 - y^3, x - y defined by a julia-function with inverse
+)
+
+julia> f
+Map from
+Multivariate Polynomial Ring in x, y over Rational Field to Quotient of Multivariate Polynomial Ring in x, y over Rational Field by ideal generated by: x^2 - y^3, x - y defined by a julia-function with inverse
+```
 """
 function quo(R::MPolyRing, I::MPolyIdeal) 
   q = MPolyQuo(R, I)

src/Rings/mpoly-affine-algebras.jl

@@ -15,6 +15,19 @@ export issurjective, isinjective, isbijective, inverse, preimage, isfinite
     dim(A::MPolyQuo)
 
 Return the dimension of `A`.
+###### Example
+```jldoctest
+julia> R, (x, y, z) = PolynomialRing(QQ, ["x", "y", "z"])
+(Multivariate Polynomial Ring in x, y, z over Rational Field, fmpq_mpoly[x, y, z])
+
+julia> A, _ = quo(R, ideal(R, [y-x^2, x-z^3]))
+(Quotient of Multivariate Polynomial Ring in x, y, z over Rational Field by ideal generated by: -x^2 + y, x - z^3, Map from
+Multivariate Polynomial Ring in x, y, z over Rational Field to Quotient of Multivariate Polynomial Ring in x, y, z over Rational Field by ideal generated by: -x^2 + y, x - z^3 defined by a julia-function with inverse
+)
+
+julia> dim(A)
+1
+```
 """
 function dim(A::MPolyQuo) 
   I = A.I

src/Rings/mpoly-ideals.jl

@@ -204,10 +204,27 @@ function saturation(I::MPolyIdeal, J::MPolyIdeal)
   return MPolyIdeal(I.gens.Ox, Singular.saturation(I.gens.S, J.gens.S))
 end
 #######################################################
+#=
+FOR LATER USE
+function saturation(I::MPolyIdeal, J::MPolyIdeal)
+  singular_assure(I)
+  singular_assure(J)
+  K, _ = Singular.saturation(I.gens.S, J.gens.S)
+  return MPolyIdeal(I.gens.Ox, K)
+ end 
+function saturation_with_index(I::MPolyIdeal, J::MPolyIdeal)
+  singular_assure(I)
+  singular_assure(J)
+  K, k = Singular.saturation(I.gens.S, J.gens.S)
+  return (MPolyIdeal(I.gens.Ox, K), k)
+ end
+ =#
+#######################################################
+
 
 # elimination #######################################################
 @doc Markdown.doc"""
-    eliminate(I::MPolyIdeal, polys::Array{MPolyElem, 1})
+    eliminate(I::MPolyIdeal, l::Array{MPolyElem, 1})
 
 Given a list of polynomials which are variables, these variables are eliminated from `I`. 
 
@@ -237,7 +254,7 @@ function eliminate(I::MPolyIdeal, l::Array{<:MPolyElem, 1})
 end
 
 @doc Markdown.doc"""
-    eliminate(I::MPolyIdeal, polys::AbstractArray{Int, 1})
+    eliminate(I::MPolyIdeal, l::AbstractArray{Int, 1})
 
 Given a list of indices which specify variables, these variables are eliminated from `I`. 
 
@@ -282,7 +299,8 @@ If the base ring of `I` is a polynomial
 ring over a field, a combination of the algorithms of Krick and Logar 
 (with modifications by Laplagne) and Kemper is used. For polynomial
 rings over the integers, the algorithm proceeds as suggested by 
-Pfister, Sadiq, and Steidel.
+Pfister, Sadiq, and Steidel. See [KL91](@cite),
+[Kem02](@cite), and [PSS11](@cite).
 ###### Examples
 ```jldoctest
 julia> R, (x, y) = PolynomialRing(QQ, ["x", "y"])
@@ -343,7 +361,7 @@ Return a primary decomposition of `I`. If `I` is the unit ideal, return `[ideal(
 If the base ring of `I` is a polynomial ring over a field, the algorithm of Gianni, Trager and Zacharias 
 is used by default. Alternatively, the algorithm by Shimoyama and Yokoyama can be used by specifying 
 `alg=:SY`.  For polynomial rings over the integers, the algorithm proceeds as suggested by 
-Pfister, Sadiq, and Steidel.
+Pfister, Sadiq, and Steidel. See [GTZ88](@cite), [SY96](@cite), and [PSS11](@cite).
 ###### Examples
 ```jldoctest
 julia> R, (x, y) = PolynomialRing(QQ, ["x", "y"])
@@ -422,6 +440,7 @@ If the base ring of `I` is a polynomial ring over a field, the algorithm of
 Gianni-Trager-Zacharias is used by default and characteristic sets may be
 used by specifying `alg=:charSets`. For polynomial rings over the integers, 
 the algorithm proceeds as suggested by Pfister, Sadiq, and Steidel.
+See [GTZ88](@cite) and [PSS11](@cite).
 ###### Examples
 ```jldoctest
 julia> R, (x, y) = PolynomialRing(QQ, ["x", "y"])
@@ -495,7 +514,7 @@ primes of `I` of that dimension. If `I` is the unit ideal, return `[ideal(1)]`.
 
 # Implemented Algorithms
 
-The implementation relies on ideas of Eisenbud, Huneke, and Vasconcelos.
+The implementation relies on ideas of Eisenbud, Huneke, and Vasconcelos. See [EHV92](@cite). 
 ###### Example
 ```jldoctest
 julia> R, (x, y) = PolynomialRing(QQ, ["x", "y"])
@@ -529,7 +548,7 @@ of `I` of that dimension. If `I` is the unit ideal, return `[ideal(1)]`.
 
 # Implemented Algorithms
 
-The implementation combines the algorithms of Krick and Logar (with modifications by Laplagne) and Kemper.
+The implementation combines the algorithms of Krick and Logar (with modifications by Laplagne) and Kemper. See [KL91](@cite) and [Kem02](@cite).
 ###### Example
 ```jldoctest
 julia> R, (x, y) = PolynomialRing(QQ, ["x", "y"])
@@ -566,7 +585,8 @@ minimal height.  If `I` is the unit ideal, return `[ideal(1)]`.
 
 For polynomial rings over a field, the implementation relies on ideas as used by
 Gianni, Trager, and Zacharias or Krick and Logar. For polynomial rings over the integers, 
-the algorithm proceeds as suggested by Pfister, Sadiq, and Steidel.
+the algorithm proceeds as suggested by Pfister, Sadiq, and Steidel. See [GTZ88](@cite),
+[KL91](@cite),  and [PSS11](@cite).
 ###### Examples
 ```jldoctest
 julia> R, (x, y) = PolynomialRing(QQ, ["x", "y"])
@@ -619,7 +639,7 @@ If `I` is the unit ideal, return `[ideal(1)]`.
 # Implemented Algorithms
 
 The implementation relies on a combination of the algorithms of Krick and Logar 
-(with modifications by Laplagne) and Kemper. 
+(with modifications by Laplagne) and Kemper. See [KL91](@cite) and [Kem02](@cite).
 ###### Example
 ```jldoctest
 julia> R, (x, y) = PolynomialRing(QQ, ["x", "y"])