Local Polynomial Rings
- Introduction
- Elements and Local Monomial Orders
- Local Polynomial Rings and Ideals
- Creation of Local Polynomial Rings and Accessing their Monomial Orders
LocalPolynomialRing(K, n): Rng, RngIntElt → RngMPolLocLocalPolynomialRing(K, n, order): Rng, RngIntElt, MonStgElt, ... → RngMPolLocLocalPolynomialAlgebra(K, n, order): Rng, RngIntElt, MonStgElt, ... → RngMPolLocLocalPolynomialRing(K, n, T): Rng, RngIntElt, Tup → RngMPolLocMonomialOrder(R): RngMPolLoc → TupMonomialOrderWeightVectors(R): RngMPol → [ [ FldRatElt ] ]Localization(R): RngMPol → RngMPolLocLocalization(I): RngMPol → RngMPolLoc- Example: Local Polynomial Rings with Different Orders
- Creation of Ideals and Accessing their Bases
ideal<R | L>: RngMPolLoc, List → RngMPolLocIdeal(B): [ RngMPolLocElt ] → RngMPolLocIdeal(B): { RngMPolLocElt } → RngMPolLocIdeal(f): RngMPolLocElt → RngMPolLocIdealWithFixedBasis(B): [ RngMPolLocElt ] → RngMPolLocBasis(I): RngMPolLoc → [ RngMPolLocElt ]BasisElement(I, i): RngMPolLoc, RngIntElt → RngMPolLocElt
- Creation of Local Polynomial Rings and Accessing their Monomial Orders
- Standard Bases
- Construction of Standard Bases
StandardBasis(I): RngMPolLoc → RngMPolLocEltStandardBasis(S): [ RngMPolLocElt ] → [ RngMPolLocElt ]StandardBasis(S): { RngMPolLocElt } → [ RngMPolLocElt ]Coordinates(I, f): RngMPolLoc, RngMPolLocElt → [ RngMPolLocElt ]CoordinateMatrix(I): RngMPolLoc → Matrix- Example: Standard Basis of an Ideal
- Example: Standard Basis Versus Groebner Basis
- Construction of Standard Bases
- Operations on Ideals
- Basic Operations
I + J: RngMPolLoc, RngMPolLoc → RngMPolLocI * J: RngMPolLoc, RngMPolLoc → RngMPolLocI ^ k: RngMPolLoc, RngIntElt → RngMPolLocQuotientDimension(I): RngMPol → RngIntEltGeneric(I): RngMPolLoc → RngMPolLocLeadingMonomialIdeal(I): RngMPolLoc → RngMPolLocI meet J: RngMPolLoc, RngMPolLoc → RngMPolLoc&meet S: [ RngMPolLoc ] → RngMPolLoc
- Ideal Predicates
I eq J: RngMPolLoc, RngMPolLoc → BoolEltI ne J: RngMPolLoc, RngMPolLoc → BoolEltI notsubset J: RngMPolLoc, RngMPolLoc → BoolEltI subset J: RngMPolLoc, RngMPolLoc → BoolEltIsZero(I): RngMPolLoc → BoolEltIsProper(I): RngMPolLoc → BoolEltIsZeroDimensional(I): RngMPolLoc → BoolElt- Example: Ideal Arithmetic in a Local Ring
- Operations on Elements of Ideals
- Basic Operations
- Changing Coefficient Ring
- Changing Monomial Order
- Dimension of Ideals
- Bibliography