More tests for gradings. Corrections.

docs/src/CommutativeAlgebra/affine_algebras.md

@@ -292,10 +292,10 @@ minimal_subalgebra_generators(V::Vector{T}) where T <: Union{MPolyElem, MPolyQuo
 ## Noether Normalization
 
 ```@docs
-noether_normalization(A::MPolyQuo)
+noether_normalization(A::MPolyQuo{<:MPolyElem{<:FieldElem}})
 ```
 
-###### Example
+###### Examples
 
 ```@repl oscar
 R, (x, y, z) = PolynomialRing(QQ, ["x", "y", "z"]);
@@ -305,32 +305,14 @@ L[1]
 L[2]
 L[3]
 ```
-
-## Normalization of Rings
+## Normalization
 
 ```@docs
-normalization(A::MPolyQuo)
-normalization_with_delta(A::MPolyQuo)
-```
-
-###### Examples
-
-```@repl oscar
-R, (x, y) = PolynomialRing(QQ, ["x", "y"])
-A, _ = quo(R, ideal(R, [(x^2-y^3)*(x^2+y^2)*x]))
-L = normalization(A)
-```
-
-```@repl oscar
-R, (x, y, z) = PolynomialRing(QQ, ["x", "y", "z"])
-A, _ = quo(R, ideal(R, [z^3-x*y^4]))
-L = normalization(A)
+normalization(A::MPolyQuo{<:MPolyElem{<:FieldElem}})
 ```
 
-```@repl oscar
-R, (x, y) = PolynomialRing(QQ, ["x", "y"])
-A, _ = quo(R, ideal(R, [(x^2-y^3)*(x^2+y^2)*x]))
-L = normalization_with_delta(A)
+```@docs
+normalization_with_delta(A::MPolyQuo{<:MPolyElem{<:FieldElem}})
 ```
 
 ## Integral Bases
@@ -339,15 +321,6 @@ L = normalization_with_delta(A)
 integral_basis(f::MPolyElem, i::Int)
 ```
 
-###### Example
-
-
-```@repl oscar
-R, (x, y) = PolynomialRing(QQ, ["x", "y"])
-f = (y^2-2)^2 + x^5
-integral_basis(f, 2)
-```
-
 ## Tests on Affine Algebras
 
 ### Reducedness Test
@@ -362,7 +335,7 @@ isreduced(Q::MPolyQuo)
 isnormal(A::MPolyQuo)
 ```
 
-###### Example
+###### Examples
 
 ```@repl oscar
 R, (x, y, z) = PolynomialRing(QQ, ["x", "y", "z"])
@@ -374,7 +347,7 @@ isnormal(A)
 
 iscohenmacaulay(R)
 
-###### Example
+###### Examples
 
 ## Hilbert Series and Hilbert Polynomial
 

docs/src/CommutativeAlgebra/rings.md

@@ -137,12 +137,14 @@ are thought of as column vectors in $\mathbb Z^m$, and $W$ as an $m \times n$-ma
 entries in $\mathbb Z$. In particular, if $G = \mathbb Z$, then $W$ is thought of as a row vector
 in $\mathbb Z^n$.
 
+We refer to the textbooks [MS05](@cite) and [KR05](@cite) for details on multigradings. With respect to notation,
+we follow the former book.
+
 !!! note
     Given a $G$-grading on $R$, we say that $R$ is *$G$-graded*, or simply that $R$  is *graded*.
-    We say that $R$ is *positively graded (by $G$)* if each graded part $R_g$ has finite rank.
-    Equivalently, the degree zero part consists of the constants only.
-
-See [MS05](@cite) and [KR05](@cite) for details on multigradings.
+    We say that $R$ is *positively graded (by $G$)* if $G$ is torsion-free and each graded part $R_g$
+	has finite rank. The latter condition is equivalent to the condition that the degree zero
+	part consists of the constants only (see Theorem 8.6 in [MS05](@cite).
 
 
 ### Types
@@ -176,6 +178,10 @@ GradedPolynomialRing(C::Ring, V::Vector{String}, W; ordering=:lex)
 
 ### Tests on Graded Rings
 
+```@docs
+is_standard_graded(R::MPolyRing_dec)
+```
+
 ```@docs
 is_z_graded(R::MPolyRing_dec)
 ```
@@ -320,4 +326,3 @@ degree(f::MPolyElem_dec)
 
 How to handle homomorphisms of multivariate polynomial rings and their graded versions is described in
 a more general context in the section on affine algebras. 
-

src/Rings/mpoly-affine-algebras.jl

@@ -4,6 +4,7 @@ export noether_normalization, normalization, integral_basis
 export isreduced, subalgebra_membership
 export hilbert_series, hilbert_series_reduced, hilbert_series_expanded, hilbert_function, hilbert_polynomial, degree
 export issurjective, isinjective, isbijective, inverse, preimage, isfinite
+export _multi_hilbert_series, _multi_hilbert_series_reduced
 
 ##############################################################################
 #
@@ -40,6 +41,10 @@ end
 #
 ##############################################################################
 
+
+
+
+
 @doc Markdown.doc"""
     hilbert_series(A::MPolyQuo)
 
@@ -199,7 +204,7 @@ return `true` if `A` is normal, `false` otherwise.
 """
 function isnormal(A::MPolyQuo)
   _, _, d = normalization_with_delta(A)
-  return d == 1
+  return d == 0
 end
 
 ##############################################################################
@@ -473,7 +478,7 @@ function _conv_normalize_data(A, l, br)
 end
 
 @doc Markdown.doc"""
-    normalization(A::MPolyQuo; alg = :equidimDec)
+    normalization(A::MPolyQuo{<:MPolyElem{<:FieldElem}}; alg = :equidimDec)
 
 Find the normalization of a reduced affine algebra over a perfect field $K$.
 That is, given the quotient $A=R/I$ of a multivariate polynomial ring $R$ over $K$
@@ -486,11 +491,11 @@ $f: A \rightarrow \overline{A}$.
 The function relies on the algorithm 
 of Greuel, Laplagne, and Seelisch which proceeds by finding a suitable decomposition 
 $I=I_1\cap\dots\cap I_r$ into radical ideals $I_k$, together with
-the normalization maps $f_k: R/I_k \rightarrow A_k=\overline{R/I_k}$, such that
+maps $A = R/I \rightarrow A_k=\overline{R/I_k}$ which give rise to the normalization map of $A$:
 
-$f=f_1\times \dots\times f_r: A \rightarrow A_1\times \dots\times A_r=\overline{A}$
+$A\hookrightarrow A_1\times \dots\times A_r=\overline{A}$
 
-is the normalization map of $A$. For each $k$, the function specifies two representations
+For each $k$, the function specifies two representations
 of $A_k$: It returns an array of triples $(A_k, f_k, \mathfrak a_k)$,
 where $A_k$ is represented as an affine $K$-algebra, and $f_k$ as a map of affine $K$-algebras.
 The third entry $\mathfrak a_k$ is a tuple $(d_k, J_k)$, consisting of an element
@@ -506,8 +511,40 @@ See [GLS10](@cite).
 
 !!! warning
     The function does not check whether $A$ is reduced. Use `isreduced(A)` in case you are unsure (this may take some time).
+
+# Examples
+```jldoctest
+julia> R, (x, y) = PolynomialRing(QQ, ["x", "y"]);
+
+julia> A, _ = quo(R, ideal(R, [(x^2-y^3)*(x^2+y^2)*x]));
+
+julia> L = normalization(A);
+
+julia> size(L)
+(2,)
+
+julia> LL = normalization(A, alg = :primeDec);
+
+julia> size(LL)
+(3,)
+
+julia> LL[1][1]
+Quotient of Multivariate Polynomial Ring in T(1), x, y over Rational Field by ideal(-T(1)*y + x, -T(1)*x + y^2, T(1)^2 - y, -x^2 + y^3)
+
+julia> LL[1][2]
+Algebra homomorphism with
+
+domain: Quotient of Multivariate Polynomial Ring in x, y over Rational Field by ideal(x^5 - x^3*y^3 + x^3*y^2 - x*y^5)
+
+codomain: Quotient of Multivariate Polynomial Ring in T(1), x, y over Rational Field by ideal(-T(1)*y + x, -T(1)*x + y^2, T(1)^2 - y, -x^2 + y^3)
+
+defining images of generators: MPolyQuoElem{fmpq_mpoly}[x, y]
+
+julia> LL[1][3]
+(y, ideal(x, y))
+```
 """
-function normalization(A::MPolyQuo; alg=:equidimDec)
+function normalization(A::MPolyQuo{<:MPolyElem{<:FieldElem}}; alg=:equidimDec)
   I = A.I
   br = base_ring(A.R)
   singular_assure(I)
@@ -516,11 +553,11 @@ function normalization(A::MPolyQuo; alg=:equidimDec)
 end
 
 @doc Markdown.doc"""
-    normalization_with_delta(A::MPolyQuo; alg = :equidimDec)
+    normalization_with_delta(A::MPolyQuo{<:MPolyElem{<:FieldElem}}; alg = :equidimDec)
 
 Compute the normalization
 
-$f=f_1\times \dots\times f_r: A \rightarrow A_1\times \dots\times A_r=\overline{A}$
+$A\hookrightarrow A_1\times \dots\times A_r=\overline{A}$
 
 of $A$ as does `normalize(A)`, but return additionally the `delta invariant` of $A$,
 that is, the dimension 
@@ -533,8 +570,43 @@ The return value is a tuple whose first element is `normalize(A)`, whose second
 containing the delta invariants of the $A_k$, and whose third element is the
 (total) delta invariant of $A$. The return value -1 in the third element
 indicates that the delta invariant is infinite.
+
+# Examples
+```jldoctest
+julia> R, (x, y) = PolynomialRing(QQ, ["x", "y"]);
+
+julia> A, _ = quo(R, ideal(R, [(x^2-y^3)*(x^2+y^2)*x]));
+
+julia> L = normalization_with_delta(A);
+
+julia> L[2]
+3-element Vector{Int64}:
+ 1
+ 1
+ 0
+
+julia> L[3]
+13
+
+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, [z^3-x*y^4]))
+(Quotient of Multivariate Polynomial Ring in x, y, z over Rational Field by ideal(-x*y^4 + 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(-x*y^4 + z^3) defined by a julia-function with inverse)
+
+julia> L = normalization_with_delta(A)
+(Tuple{MPolyQuo{fmpq_mpoly}, AlgHom{fmpq}, Tuple{MPolyQuoElem{fmpq_mpoly}, MPolyQuoIdeal{fmpq_mpoly}}}[(Quotient of Multivariate Polynomial Ring in T(1), T(2), x, y, z over Rational Field by ideal(T(1)*y - T(2)*z, T(2)*y - z, -T(1)*z + x*y^2, T(1)^2 - x*z, T(1)*T(2) - x*y, -T(1) + T(2)^2, x*y^4 - z^3), Algebra homomorphism with
+
+domain: Quotient of Multivariate Polynomial Ring in x, y, z over Rational Field by ideal(-x*y^4 + z^3)
+
+codomain: Quotient of Multivariate Polynomial Ring in T(1), T(2), x, y, z over Rational Field by ideal(T(1)*y - T(2)*z, T(2)*y - z, -T(1)*z + x*y^2, T(1)^2 - x*z, T(1)*T(2) - x*y, -T(1) + T(2)^2, x*y^4 - z^3)
+
+defining images of generators: MPolyQuoElem{fmpq_mpoly}[x, y, z]
+, (z^2, ideal(x*y^2*z, x*y^3, z^2)))], [-1], -1)
+```
 """
-function normalization_with_delta(A::MPolyQuo; alg=:equidimDec)
+function normalization_with_delta(A::MPolyQuo{<:MPolyElem{<:FieldElem}}; alg=:equidimDec)
   I = A.I
   br = base_ring(A.R)
   singular_assure(I)
@@ -550,18 +622,19 @@ end
 ##############################################################################
 
 @doc Markdown.doc"""
-    noether_normalization(A::MPolyQuo)
+    noether_normalization(A::MPolyQuo{<:MPolyElem{<:FieldElem}})
 
-Given an affine algebra $A=R/I$ over a field $K$, the return value is a triple $(V,F,G)$ such that:
-$V$ is a vector of $d=\dim A$ elements $l_i$ of $A$, all represented by linear forms in $R$, and
-such that $K[V]\rightarrow A$ is a Noether normalization for $A$; $F: A \rightarrow A$ is an
-automorphism of $A$, induced by a linear change of coordinates of $R$, and mapping the
-$l_i$ to the last $d$ variables of $A$; and $G = F^{-1}$.
+Given an affine algebra $A=R/I$ over a field $K$, return a triple $(V,F,G)$ such that:
+$V$ is a vector of $d=\dim A$ elements of $A$, represented by linear forms $l_i\in R$, and
+such that $K[V]\hookrightarrow A$ is a Noether normalization for $A$; $F: A=R/I \rightarrow B = R/\phi(I)$ 
+is an isomorphism, induced by a linear change $ \phi $ of coordinates of $R$ which maps the
+$l_i$ to the the last $d$ variables of $R$; and $G = F^{-1}$.
 
 !!! warning
     The algorithm may not terminate over a small finite field. If it terminates, the result is correct.
+
 """
-function noether_normalization(A::MPolyQuo)
+function noether_normalization(A::MPolyQuo{<:MPolyElem{<:FieldElem}})
  I = A.I
  R = base_ring(I)
  singular_assure(I)
@@ -570,13 +643,18 @@ function noether_normalization(A::MPolyQuo)
  i2 = [R(x) for x = gens(l[2])]
  m = matrix([[coeff(x, y) for y = gens(R)] for x = i1])
  mi = inv(m)
- mi_arr = [collect(matrix([gens(R)])*map_entries(R, mi))[i] for i in 1:ngens(R)]
- h1 = AlgebraHomomorphism(A, A, map(A, i1))
- h2 = AlgebraHomomorphism(A, A, map(A, mi_arr))
- return map(x->h2(A(x)), i2), h1, h2
+ ###mi_arr = [collect(matrix([gens(R)])*map_entries(R, mi))[i] for i in 1:ngens(R)]
+ mi_arr = [collect(map_entries(R, mi)*gens(R))[i] for i in 1:ngens(R)]
+ h = AlgebraHomomorphism(R, R, i1)
+ V = map(x->h(x), gens(I))
+ B, _ = quo(R, ideal(R, V))
+ h1 = AlgebraHomomorphism(A, B, map(B, i1))
+ h2 = AlgebraHomomorphism(B, A, map(A, mi_arr))
+ return map(x->h2(B(x)), i2), h1, h2
 end
 
 
+
 ##############################################################################
 #
 # Integral bases
@@ -593,27 +671,43 @@ Then the normalization of $A = Q[x,y]/\langle f \rangle$, that is, the
 integral closure $\overline{A}$ of $A$ in its quotient field, is a free
 module over $K[x]$ of finite rank, and any set of free generators for
 $\overline{A}$ over $K[x]$ is called an *integral basis* for $\overline{A}$
-over $K[x]$. Relying on the algorithm by Böhm, Decker, Laplagne, and Pfister,
+over $K[x]$. Relying on the algorithm by [BDLP19](@cite),
 the function returns a pair $(d, V)$, where $d$ is an element of $A$,
 and $V$ is a vector of elements in $A$, such that the fractions $v/d, v\in V$,
-form an integral basis for $\overline{A}$ over $K[x]$. See [BDLP19](@cite).
+form an integral basis for $\overline{A}$ over $K[x]$. 
 
 !!! note
     The conditions on $f$ are automatically checked.
+
+# Examples
+```jldoctest
+julia> R, (x, y) = PolynomialRing(QQ, ["x", "y"])
+(Multivariate Polynomial Ring in x, y over Rational Field, fmpq_mpoly[x, y])
+
+julia> f = (y^2-2)^2 + x^5
+x^5 + y^4 - 4*y^2 + 4
+
+julia> integral_basis(f, 2)
+(x^2, MPolyQuoElem{fmpq_mpoly}[x^2, x^2*y, y^2 - 2, y^3 - 2*y])
+```
 """
 function integral_basis(f::MPolyElem, i::Int)
   R = parent(f)
 
+  if typeof(R) <: MPolyRing_dec
+    throw(ArgumentError("Not implemented for decorated rings."))
+  end
+
   if !(nvars(R) == 2)
-    throw(ArgumentError("The base ring must be a ring in two variables."))
+    throw(ArgumentError("The parent ring must be a polynomial ring in two variables."))
   end
 
   if !(i == 1 || i == 2)
     throw(ArgumentError("The index $i must be either 1 or 2, indicating the integral variable."))
   end
 
-  if !(base_ring(R) == QQ || base_ring(R) == Singular.QQ)
-    throw(ArgumentError("The base ring must be the rationals."))
+  if !(coefficient_ring(R) == QQ || base_ring(R) == Singular.QQ)
+    throw(ArgumentError("The coefficient ring must be the rationals."))
   end
 
   if !isone(coeff(f, [i], [degree(f, i)]))
@@ -626,6 +720,8 @@ function integral_basis(f::MPolyElem, i::Int)
 
   SR = singular_ring(R)
   l = Singular.LibIntegralbasis.integralBasis(SR(f), i, "isIrred")
-  return (R(l[2]), R.(gens(l[1])))
+  A, p = quo(R, ideal(R, [f]))
+  ###return (R(l[2]), R.(gens(l[1])))
+  return (p(R(l[2])), [p(R(x)) for x = gens(l[1])])
 end