next
|
previous
|
forward
|
backward
| up |
top
|
index
|
toc
|
Macaulay2 web site
Macaulay2Doc
::
RingElement
RingElement -- the class of all ring elements handled by the engine
See also
engine
Functions and methods returning a ring element :
RingElement * RingElement, see
*
-- a binary operator, usually used for multiplication
RingElement + RingElement, see
+
-- a unary or binary operator, usually used for addition
- RingElement, see
-
-- a unary or binary operator, usually used for negation or subtraction
RingElement - RingElement, see
-
-- a unary or binary operator, usually used for negation or subtraction
chi(CoherentSheaf)
(missing documentation) -- compute the Euler characteristic of a coherent sheaf
contract(Number,RingElement), see
contract(Matrix,Matrix)
-- contract a matrix by a matrix
contract(RingElement,Number), see
contract(Matrix,Matrix)
-- contract a matrix by a matrix
contract(RingElement,RingElement), see
contract(Matrix,Matrix)
-- contract a matrix by a matrix
determinant(Matrix), see
determinant
-- determinant of a matrix
diff(RingElement,RingElement)
-- differentiation
exp(RingElement)
gcd(RingElement,RingElement), see
gcd
-- greatest common divisor
generator(Ideal), see
generator
-- provide a single generator
generator(Module), see
generator
-- provide a single generator
Ideal _ ZZ, see
generators of ideals and modules
homogenize(RingElement,RingElement,List), see
homogenize
-- homogenize with respect to a variable
IndexedVariable _ Ring
-- get a ring variable by name
leadCoefficient(RingElement), see
leadCoefficient
-- the coefficient of the leading term
leadMonomial(RingElement), see
leadMonomial
-- the leading monomial of a ring element
leadTerm(RingElement)
-- get the greatest term
leadTerm(ZZ,RingElement)
-- get the lead polynomials using part of the monomial order
RingElement % GroebnerBasis, see
Matrix % GroebnerBasis
-- calculate the normal form of ring elements and matrices using a (partially computed) Gröbner basis
RingElement // RingElement, see
Matrix // Matrix
-- factor a map through another
Matrix _ Sequence
-- get entry of matrix
RingElement % RingElement, see
methods for normal forms and remainder
-- normal form of ring elements and matrices
part(InfiniteNumber,InfiniteNumber,RingElement), see
part
-- select terms of a polynomial by degree(s) or weight(s)
part(InfiniteNumber,InfiniteNumber,VisibleList,RingElement), see
part
-- select terms of a polynomial by degree(s) or weight(s)
part(InfiniteNumber,ZZ,RingElement), see
part
-- select terms of a polynomial by degree(s) or weight(s)
part(InfiniteNumber,ZZ,VisibleList,RingElement), see
part
-- select terms of a polynomial by degree(s) or weight(s)
part(List,RingElement), see
part
-- select terms of a polynomial by degree(s) or weight(s)
part(Nothing,Nothing,RingElement), see
part
-- select terms of a polynomial by degree(s) or weight(s)
part(Nothing,Nothing,VisibleList,RingElement), see
part
-- select terms of a polynomial by degree(s) or weight(s)
part(Nothing,ZZ,RingElement), see
part
-- select terms of a polynomial by degree(s) or weight(s)
part(Nothing,ZZ,VisibleList,RingElement), see
part
-- select terms of a polynomial by degree(s) or weight(s)
part(ZZ,InfiniteNumber,RingElement), see
part
-- select terms of a polynomial by degree(s) or weight(s)
part(ZZ,InfiniteNumber,VisibleList,RingElement), see
part
-- select terms of a polynomial by degree(s) or weight(s)
part(ZZ,Nothing,RingElement), see
part
-- select terms of a polynomial by degree(s) or weight(s)
part(ZZ,Nothing,VisibleList,RingElement), see
part
-- select terms of a polynomial by degree(s) or weight(s)
part(ZZ,RingElement), see
part
-- select terms of a polynomial by degree(s) or weight(s)
part(ZZ,VisibleList,RingElement), see
part
-- select terms of a polynomial by degree(s) or weight(s)
part(ZZ,ZZ,RingElement), see
part
-- select terms of a polynomial by degree(s) or weight(s)
part(ZZ,ZZ,VisibleList,RingElement), see
part
-- select terms of a polynomial by degree(s) or weight(s)
poly(String)
-- make a polynomial using classic Macaulay syntax
pseudoRemainder(RingElement,RingElement), see
pseudoRemainder
-- compute the pseudo-remainder
random(List,Ring), see
random(Type)
-- random element of a type
random(ZZ,Ring), see
random(Type)
-- random element of a type
Ring _ List
-- make a monomial from a list of exponents
Ring _ String
-- get a ring variable by name
Ring _ ZZ
-- get a ring variable by index
RingElement / RingElement
-- fraction
RingElement ^ ZZ
-- power
RingMap RingElement
-- apply a ring map
round(ZZ,RingElement)
(missing documentation)
someTerms(RingElement,ZZ,ZZ), see
someTerms
-- select some terms of a polynomial
substitute(Number,Ring), see
substitute
-- substituting values for variables
substitute(Number,RingFamily), see
substitute
-- substituting values for variables
substitute(RingElement,List), see
substitute
-- substituting values for variables
substitute(RingElement,Matrix), see
substitute
-- substituting values for variables
substitute(RingElement,Ring), see
substitute
-- substituting values for variables
substitute(RingElement,RingFamily), see
substitute
-- substituting values for variables
Symbol _ Ring
-- get a ring variable by name
trace(Matrix)
-- trace of a matrix
Methods that use a ring element :
Number % RingElement, see
%
-- a binary operator, usually used for remainder and reduction
RingElement % Number, see
%
-- a binary operator, usually used for remainder and reduction
Matrix * RingElement, see
*
-- a binary operator, usually used for multiplication
Ring * RingElement, see
*
-- a binary operator, usually used for multiplication
RingElement * ChainComplexMap, see
*
-- a binary operator, usually used for multiplication
RingElement * GradedModuleMap, see
*
-- a binary operator, usually used for multiplication
RingElement * Ideal, see
*
-- a binary operator, usually used for multiplication
RingElement * Matrix, see
*
-- a binary operator, usually used for multiplication
RingElement * Module, see
*
-- a binary operator, usually used for multiplication
RingElement * MonomialIdeal, see
*
-- a binary operator, usually used for multiplication
RingElement * MutableMatrix, see
*
-- a binary operator, usually used for multiplication
RingElement * Vector, see
*
-- a binary operator, usually used for multiplication
ChainComplexMap + RingElement, see
+
-- a unary or binary operator, usually used for addition
GradedModuleMap + RingElement, see
+
-- a unary or binary operator, usually used for addition
Ideal + RingElement, see
+
-- a unary or binary operator, usually used for addition
Matrix + RingElement, see
+
-- a unary or binary operator, usually used for addition
RingElement + ChainComplexMap, see
+
-- a unary or binary operator, usually used for addition
RingElement + GradedModuleMap, see
+
-- a unary or binary operator, usually used for addition
RingElement + Matrix, see
+
-- a unary or binary operator, usually used for addition
+ RingElement
(missing documentation)
ChainComplexMap - RingElement, see
-
-- a unary or binary operator, usually used for negation or subtraction
GradedModuleMap - RingElement, see
-
-- a unary or binary operator, usually used for negation or subtraction
Matrix - RingElement, see
-
-- a unary or binary operator, usually used for negation or subtraction
RingElement - ChainComplexMap, see
-
-- a unary or binary operator, usually used for negation or subtraction
RingElement - GradedModuleMap, see
-
-- a unary or binary operator, usually used for negation or subtraction
RingElement - Matrix, see
-
-- a unary or binary operator, usually used for negation or subtraction
Number // RingElement, see
//
-- a binary operator, usually used for quotient
RingElement // Number, see
//
-- a binary operator, usually used for quotient
ChainComplexMap == RingElement, see
==
-- equality
GradedModuleMap == RingElement, see
==
-- equality
Matrix == RingElement, see
==
-- equality
Number == RingElement, see
==
-- equality
RingElement == ChainComplexMap, see
==
-- equality
RingElement == GradedModuleMap, see
==
-- equality
RingElement == Matrix, see
==
-- equality
RingElement == Number, see
==
-- equality
RingElement == RingElement, see
==
-- equality
RingElement == ZZ, see
==
-- equality
ZZ == RingElement, see
==
-- equality
analyticSpread(Ideal,RingElement), see
analyticSpread
-- Compute the analytic spread of a module or ideal
analyticSpread(Module,RingElement), see
analyticSpread
-- Compute the analytic spread of a module or ideal
annihilator(RingElement), see
annihilator
-- the annihilator ideal
antipode(RingElement), see
antipode
-- antipode for skew commuting polynomial rings
baseName(RingElement), see
baseName
-- the base name of a generator
binomial(RingElement,ZZ)
(missing documentation)
clean(RR,RingElement), see
clean
-- Set to zero elements that are approximately zero
coefficients(RingElement), see
coefficients
-- monomials and their coefficients
cokernel(RingElement), see
cokernel
-- cokernel of a map of modules, graded modules, or chaincomplexes
columnAdd(MutableMatrix,ZZ,RingElement,ZZ), see
columnAdd
-- add a multiple of one column to another
columnMult(MutableMatrix,ZZ,RingElement), see
columnMult
-- multiply a column by a ring element
Constant * RingElement, see
Constant
Constant + RingElement, see
Constant
Constant - RingElement, see
Constant
Constant / RingElement, see
Constant
RingElement * Constant, see
Constant
RingElement + Constant, see
Constant
RingElement - Constant, see
Constant
RingElement / Constant, see
Constant
content(RingElement)
-- the content of a polynomial
content(RingElement,RingElement)
(missing documentation)
contract(Matrix,RingElement), see
contract(Matrix,Matrix)
-- contract a matrix by a matrix
contract(RingElement,Matrix), see
contract(Matrix,Matrix)
-- contract a matrix by a matrix
contract(RingElement,Vector), see
contract(Matrix,Matrix)
-- contract a matrix by a matrix
contract(Vector,RingElement), see
contract(Matrix,Matrix)
-- contract a matrix by a matrix
degree(RingElement)
degree(RingElement,RingElement)
-- degree with respect to a variable
diff(RingElement,Vector), see
diff(Matrix,Matrix)
-- differentiate a matrix by a matrix
diff(Vector,RingElement), see
diff(Matrix,Matrix)
-- differentiate a matrix by a matrix
diff(Matrix,RingElement)
-- differentiation
diff(RingElement,Matrix)
-- differentiate each entry of a matrix
discriminant(RingElement,RingElement)
divideByVariable(Matrix,RingElement), see
divideByVariable
-- divide all columns by a (power of a) variable
divideByVariable(Matrix,RingElement,ZZ), see
divideByVariable
-- divide all columns by a (power of a) variable
dual(MonomialIdeal,RingElement)
-- the Alexander dual
eliminate(RingElement,Ideal), see
eliminate
exponents(RingElement), see
exponents
-- list the exponents in a polynomial
factor(RingElement)
-- factor a ring element
fraction(RingElement,RingElement), see
fraction
fromDividedPowers(RingElement), see
fromDividedPowers
-- Translates from divided power monomial basis to ordinary monomial basis
fromDual(RingElement), see
fromDual
-- Ideal from inverse system
gcd(RingElement,ZZ), see
gcd
-- greatest common divisor
gcd(ZZ,RingElement), see
gcd
-- greatest common divisor
gcdCoefficients(RingElement,RingElement), see
gcdCoefficients
-- gcd with coefficients
genericMatrix(Ring,RingElement,ZZ,ZZ), see
genericMatrix
-- make a generic matrix of variables
genericSkewMatrix(Ring,RingElement,ZZ), see
genericSkewMatrix
-- make a generic skew symmetric matrix of variables
genericSymmetricMatrix(Ring,RingElement,ZZ), see
genericSymmetricMatrix
-- make a generic symmetric matrix
homogenize(Ideal,RingElement), see
homogenize
-- homogenize with respect to a variable
homogenize(Matrix,RingElement), see
homogenize
-- homogenize with respect to a variable
homogenize(Matrix,RingElement,List), see
homogenize
-- homogenize with respect to a variable
homogenize(Module,RingElement), see
homogenize
-- homogenize with respect to a variable
homogenize(Module,RingElement,List), see
homogenize
-- homogenize with respect to a variable
homogenize(RingElement,RingElement), see
homogenize
-- homogenize with respect to a variable
homogenize(Vector,RingElement), see
homogenize
-- homogenize with respect to a variable
homogenize(Vector,RingElement,List), see
homogenize
-- homogenize with respect to a variable
icPIdeal(RingElement,RingElement,ZZ), see
icPIdeal
-- compute the integral closure in prime characteristic of a principal ideal
Ideal * RingElement
(missing documentation)
ideal(RingElement)
-- make an ideal
idealizer(Ideal,RingElement), see
idealizer
-- compute Hom(I,I) as a quotient ring
image(RingElement), see
image
-- image of a map
index(RingElement), see
index
-- numeric index of a ring variable
indices(RingElement)
-- indices of variables occurring in a polynomial
InexactNumber % RingElement
(missing documentation)
InexactNumber * RingElement
(missing documentation)
InexactNumber + RingElement
(missing documentation)
InexactNumber - RingElement
(missing documentation)
InexactNumber / RingElement
(missing documentation)
InexactNumber // RingElement
(missing documentation)
InexactNumber == RingElement
(missing documentation)
installHilbertFunction(Ideal,RingElement), see
installHilbertFunction
-- install a Hilbert function without computation
installHilbertFunction(Matrix,RingElement), see
installHilbertFunction
-- install a Hilbert function without computation
installHilbertFunction(Module,RingElement), see
installHilbertFunction
-- install a Hilbert function without computation
integralClosure(Ideal,RingElement), see
integralClosure(Ideal,RingElement,ZZ)
-- integral closure of an ideal in an affine domain
integralClosure(Ideal,RingElement,ZZ)
-- integral closure of an ideal in an affine domain
inverseSystem(RingElement), see
inverseSystem
-- Inverse systems with equivariance
isConstant(RingElement), see
isConstant
-- whether a ring element is constant
isHomogeneous(RingElement), see
isHomogeneous
-- whether something is homogeneous (graded)
isLinearType(Ideal,RingElement), see
isLinearType
-- Determine whether module has linear type
isLinearType(Module,RingElement), see
isLinearType
-- Determine whether module has linear type
isReduction(Ideal,Ideal,RingElement), see
isReduction
-- Determine whether an ideal is a reduction
isReduction(Module,Module,RingElement), see
isReduction
-- Determine whether an ideal is a reduction
isUnit(RingElement), see
isUnit
-- whether a ring element is a unit
jacobian(RingElement)
(missing documentation)
kernel(RingElement), see
kernel(Matrix)
-- kernel of a matrix
lcm(RingElement,RingElement), see
lcm
-- least common multiple
lcm(RingElement,ZZ), see
lcm
-- least common multiple
lcm(ZZ,RingElement), see
lcm
-- least common multiple
lift(Ideal,type of RingElement), see
lift
-- lift to another ring
lift(Matrix,type of RingElement), see
lift
-- lift to another ring
RingElement ^ Ring, see
lift
-- lift to another ring
RingElement ^ RingFamily, see
lift
-- lift to another ring
lift(Module,type of RingElement)
(missing documentation)
lift(MutableMatrix,type of RingElement)
(missing documentation)
List % RingElement
(missing documentation)
List // RingElement
(missing documentation)
listForm(RingElement), see
listForm
-- convert to list form
map(Module,Module,RingElement)
-- construct the map induced by multiplication by a ring element on the generators
Matrix ** RingElement
-- a binary operator, usually used for tensor product or Cartesian product
Matrix ++ RingElement, see
Matrix ++ Matrix
-- direct sum of maps
RingElement ++ Matrix, see
Matrix ++ Matrix
-- direct sum of maps
RingElement ++ RingElement, see
Matrix ++ Matrix
-- direct sum of maps
Matrix // RingElement, see
Matrix // Matrix
-- factor a map through another
Matrix \\ RingElement, see
Matrix // Matrix
-- factor a map through another
RingElement // GroebnerBasis, see
Matrix // Matrix
-- factor a map through another
RingElement // Matrix, see
Matrix // Matrix
-- factor a map through another
RingElement // MonomialIdeal, see
Matrix // Matrix
-- factor a map through another
RingElement \\ Matrix, see
Matrix // Matrix
-- factor a map through another
Matrix | RingElement, see
Matrix | Matrix
-- join matrices horizontally
RingElement | Matrix, see
Matrix | Matrix
-- join matrices horizontally
RingElement | RingElement, see
Matrix | Matrix
-- join matrices horizontally
Matrix || RingElement, see
Matrix || Matrix
-- join matrices vertically
RingElement || Matrix, see
Matrix || Matrix
-- join matrices vertically
RingElement || RingElement, see
Matrix || Matrix
-- join matrices vertically
matrix(RingElement)
-- make a matrix from a ring element
member(RingElement,Ideal)
(missing documentation)
Matrix % RingElement, see
methods for normal forms and remainder
-- normal form of ring elements and matrices
RingElement % Ideal, see
methods for normal forms and remainder
-- normal form of ring elements and matrices
RingElement % Matrix, see
methods for normal forms and remainder
-- normal form of ring elements and matrices
RingElement % MonomialIdeal, see
methods for normal forms and remainder
-- normal form of ring elements and matrices
Module * RingElement
(missing documentation)
Module / RingElement, see
Module / Module
-- quotient module
MonomialIdeal : RingElement
(missing documentation)
monomialIdeal(RingElement), see
monomialIdeal(Matrix)
-- monomial ideal of lead monomials
monomials(RingElement), see
monomials
-- matrix of monomials in a ring element or matrix
multiplicity(Ideal,RingElement), see
multiplicity
-- Compute the Hilbert-Samuel multiplicity of an ideal
MutableMatrix * RingElement
(missing documentation)
norm(InfiniteNumber,RingElement), see
norm
norm(RingElement), see
norm
norm(RR,RingElement), see
norm
normalCone(Ideal,RingElement), see
normalCone
-- The normal cone of a subscheme
parts(RingElement), see
parts
-- display terms of a polynomial degree by degree
precision(RingElement), see
precision
projectiveHilbertPolynomial(RingElement)
(missing documentation)
RingElement _ Ring, see
promote
-- promote to another ring
promote(Module,type of RingElement)
(missing documentation)
promote(MutableMatrix,type of RingElement)
(missing documentation)
promote(Vector,type of RingElement)
(missing documentation)
pseudoRemainder(RingElement,RingElement,RingElement)
(missing documentation) -- pseudo-remainder
Ideal : RingElement, see
quotient(Ideal,Ideal)
-- ideal or submodule quotient
Module : RingElement, see
quotient(Ideal,Ideal)
-- ideal or submodule quotient
quotient(Ideal,RingElement), see
quotient(Ideal,Ideal)
-- ideal or submodule quotient
quotient(Module,RingElement), see
quotient(Ideal,Ideal)
-- ideal or submodule quotient
quotient(MonomialIdeal,RingElement)
(missing documentation)
quotientRemainder(InexactNumber,RingElement)
(missing documentation)
quotientRemainder(RingElement,InexactNumber)
(missing documentation)
quotientRemainder(Number,RingElement), see
quotientRemainder(RingElement,RingElement)
-- quotient and remainder
quotientRemainder(RingElement,Number), see
quotientRemainder(RingElement,RingElement)
-- quotient and remainder
quotientRemainder(RingElement,RingElement)
-- quotient and remainder
reesAlgebra(Ideal,RingElement), see
reesAlgebra
-- Compute the defining ideal of the Rees Algebra
reesAlgebra(Module,RingElement), see
reesAlgebra
-- Compute the defining ideal of the Rees Algebra
reesIdeal(Ideal,RingElement), see
reesIdeal
-- Compute the defining ideal of the Rees Algebra
reesIdeal(Module,RingElement), see
reesIdeal
-- Compute the defining ideal of the Rees Algebra
resultant(RingElement,RingElement,RingElement)
ring(RingElement), see
ring
-- get the associated ring of an object
Ring / RingElement, see
Ring / Ideal
-- make a quotient ring
RingElement % InexactNumber
(missing documentation)
RingElement * InexactNumber
(missing documentation)
RingElement + Ideal
(missing documentation)
RingElement + InexactNumber
(missing documentation)
RingElement - InexactNumber
(missing documentation)
RingElement .. RingElement
-- a sequence of consecutive generators of a polynomial ring
RingElement .. Thing
(missing documentation)
RingElement ..< RingElement
-- a sequence of consecutive generators of a polynomial ring
RingElement ..< Thing
(missing documentation)
RingElement / InexactNumber
(missing documentation)
RingElement // InexactNumber
(missing documentation)
RingElement == InexactNumber
(missing documentation)
RingElement ~
(missing documentation)
RingElement Array
-- substitution of variables
ringFromFractions(Matrix,RingElement), see
ringFromFractions
-- find presentation for f.g. ring
roots(RingElement)
-- compute the roots of a polynomial
rowAdd(MutableMatrix,ZZ,RingElement,ZZ), see
rowAdd
-- add a multiple of one row to another
rowMult(MutableMatrix,ZZ,RingElement), see
rowMult
-- multiply a row by a ring element
saturate(Ideal,RingElement), see
saturate
-- saturation of ideal or submodule
saturate(Module,RingElement), see
saturate
-- saturation of ideal or submodule
saturate(MonomialIdeal,RingElement), see
saturate
-- saturation of ideal or submodule
saturate(Vector,RingElement)
(missing documentation)
size(RingElement), see
size
-- the size of an object
specialFiber(Ideal,RingElement), see
specialFiber
-- Special fiber of a blowup
specialFiber(Module,RingElement), see
specialFiber
-- Special fiber of a blowup
specialFiberIdeal(Ideal,RingElement), see
specialFiberIdeal
-- Special fiber of a blowup
specialFiberIdeal(Module,RingElement), see
specialFiberIdeal
-- Special fiber of a blowup
standardForm(RingElement), see
standardForm
-- convert to standard form
substitute(RingElement,Option), see
substitute
-- substituting values for variables
support(RingElement), see
support
-- list of variables occurring in a polynomial or matrix
switch(RingElement)
(missing documentation)
sylvesterMatrix(RingElement,RingElement,RingElement)
terms(Ring,RingElement), see
terms
-- provide a list of terms of a polynomial
terms(RingElement), see
terms
-- provide a list of terms of a polynomial
testHunekeQuestion(RingElement), see
testHunekeQuestion
-- tests a conjecture on integral closures strengthening the Eisenbud-Mazur conjecture
Thing .. RingElement
(missing documentation)
Thing ..< RingElement
(missing documentation)
toDividedPowers(RingElement), see
toDividedPowers
-- Translates to divided power monomial basis from ordinary monomial basis
topCoefficients(RingElement), see
topCoefficients
-- first variable and its coefficient of a polynomial or matrix
variety(RingElement), see
variety(Ring)
-- the variety previously associated to a given ring
weightRange(List,RingElement), see
weightRange
-- the pair of lowest and highest weights of the monomials
weightRange(RingElement), see
weightRange
-- the pair of lowest and highest weights of the monomials
width(RingElement)
(missing documentation)
For the programmer
The object
RingElement
is
a
type
, with ancestor classes
BasicList
<
Thing
.