# Local Polynomial Rings

- [Introduction](introduction.md)

- [Elements and Local Monomial Orders](elements-order.md)

  - [Local Lexicographical: llex](elements-order.md#local-lexicographical-llex)

  - [Local Graded Lexicographical: lglex](elements-order.md#local-graded-lexicographical-lglex)

  - [Local Graded Reverse Lexicographical: lgrevlex](elements-order.md#local-graded-reverse-lexicographical-lgrevlex)

- [Local Polynomial Rings and Ideals](creation.md)

  - [Creation of Local Polynomial Rings and Accessing their Monomial Orders](creation.md#creation-of-local-polynomial-rings-and-accessing-their-monomial-orders)

    - [`LocalPolynomialRing(K, n): Rng, RngIntElt → RngMPolLoc`](creation.md#function-rngmpolloc-localpolynomialring)

    - [`LocalPolynomialRing(K, n, order): Rng, RngIntElt, MonStgElt, ... → RngMPolLoc`](creation.md#function-localpolynomialring-rng-rngintelt-monstgelt)

    - [`LocalPolynomialAlgebra(K, n, order): Rng, RngIntElt, MonStgElt, ... → RngMPolLoc`](creation.md#function-localpolynomialalgebra-rng-rngintelt-monstgelt)

    - [`LocalPolynomialRing(K, n, T): Rng, RngIntElt, Tup → RngMPolLoc`](creation.md#function-rngmpolloc-localpolynomialringtup)

    - [`MonomialOrder(R): RngMPolLoc → Tup`](creation.md#function-rngmpolloc-monomialorder)

    - [`MonomialOrderWeightVectors(R): RngMPol → [ [ FldRatElt ] ]`](creation.md#function-rngmpolloc-monomialorderweightvectors)

    - [`Localization(R): RngMPol → RngMPolLoc`](creation.md#function-localization-rngmpol)

    - [`Localization(I): RngMPol → RngMPolLoc`](creation.md#function-localization-rngmpol-2)

    - [`Example: Order`](creation.md#example-ex-ab71f5)

  - [Creation of Ideals and Accessing their Bases](creation.md#creation-of-ideals-and-accessing-their-bases)

    - [`ideal<R | L>: RngMPolLoc, List → RngMPolLoc`](creation.md#constructor-rngmpolloc-ideal)

    - [`Ideal(B): [ RngMPolLocElt ] → RngMPolLoc`](creation.md#function-rngmpolloc-ideal)

    - [`Ideal(B): { RngMPolLocElt } → RngMPolLoc`](creation.md#function-rngmpolloc-ideal-2)

    - [`Ideal(f): RngMPolLocElt → RngMPolLoc`](creation.md#function-rngmpolloc-idealofelt)

    - [`IdealWithFixedBasis(B): [ RngMPolLocElt ] → RngMPolLoc`](creation.md#function-rngmpolloc-idealwithfixedbasis)

    - [`Basis(I): RngMPolLoc → [ RngMPolLocElt ]`](creation.md#function-basis-rngmpolloc)

    - [`BasisElement(I, i): RngMPolLoc, RngIntElt → RngMPolLocElt`](creation.md#function-basiselement-rngmpolloc-rngintelt)

- [Standard Bases](groebner.md)

  - [Construction of Standard Bases](groebner.md#construction-of-standard-bases)

    - [`StandardBasis(I): RngMPolLoc → RngMPolLocElt`](groebner.md#function-rngmpolloc-standardbasis)

    - [`StandardBasis(S): [ RngMPolLocElt ] → [ RngMPolLocElt ]`](groebner.md#function-standardbasis-rngmpollocelt)

    - [`StandardBasis(S): { RngMPolLocElt } → [ RngMPolLocElt ]`](groebner.md#function-standardbasis-rngmpollocelt-2)

    - [`Coordinates(I, f): RngMPolLoc, RngMPolLocElt → [ RngMPolLocElt ]`](groebner.md#function-rngmpolloc-coordinates)

    - [`CoordinateMatrix(I): RngMPolLoc → Matrix`](groebner.md#function-rngmpolloc-coordinatematrix)

    - [`Example: Standard Basis`](groebner.md#example-ex-37d3dd)

    - [`Example: Standard Basis2`](groebner.md#example-ex-47561b)

- [Operations on Ideals](ideal-operation.md)

  - [Basic Operations](ideal-operation.md#basic-operations)

    - [`I + J: RngMPolLoc, RngMPolLoc → RngMPolLoc`](ideal-operation.md#operation-op-plus-rngmpolloc-rngmpolloc)

    - [`I * J: RngMPolLoc, RngMPolLoc → RngMPolLoc`](ideal-operation.md#operation-op-times-rngmpolloc-rngmpolloc)

    - [`I ^ k: RngMPolLoc, RngIntElt → RngMPolLoc`](ideal-operation.md#operation-op-pow-rngmpolloc-rngintelt)

    - [`QuotientDimension(I): RngMPol → RngIntElt`](ideal-operation.md#function-quotientdimension-rngmpol)

    - [`Generic(I): RngMPolLoc → RngMPolLoc`](ideal-operation.md#function-generic-rngmpolloc)

    - [`LeadingMonomialIdeal(I): RngMPolLoc → RngMPolLoc`](ideal-operation.md#function-leadingmonomialideal-rngmpolloc)

    - [`I meet J: RngMPolLoc, RngMPolLoc → RngMPolLoc`](ideal-operation.md#operation-op-meet-rngmpolloc-rngmpolloc)

    - [`&meet S: [ RngMPolLoc ] → RngMPolLoc`](ideal-operation.md#operation-operation-meet-rngmpolloc-rngmpolloc)

  - [Ideal Predicates](ideal-operation.md#ideal-predicates)

    - [`I eq J: RngMPolLoc, RngMPolLoc → BoolElt`](ideal-operation.md#operation-op-eq-rngmpolloc-rngmpolloc)

    - [`I ne J: RngMPolLoc, RngMPolLoc → BoolElt`](ideal-operation.md#operation-op-ne-rngmpolloc-rngmpolloc)

    - [`I notsubset J: RngMPolLoc, RngMPolLoc → BoolElt`](ideal-operation.md#operation-operation-notsubset-rngmpolloc-rngmpolloc-boolelt)

    - [`I subset J: RngMPolLoc, RngMPolLoc → BoolElt`](ideal-operation.md#operation-op-subset-rngmpolloc-rngmpolloc)

    - [`IsZero(I): RngMPolLoc → BoolElt`](ideal-operation.md#function-iszero-rngmpolloc)

    - [`IsProper(I): RngMPolLoc → BoolElt`](ideal-operation.md#function-isproper-rngmpolloc)

    - [`IsZeroDimensional(I): RngMPolLoc → BoolElt`](ideal-operation.md#function-iszerodimensional-rngmpolloc)

    - [`Example: Ideal Arithmetic`](ideal-operation.md#example-ex-b95527)

  - [Operations on Elements of Ideals](ideal-operation.md#operations-on-elements-of-ideals)

    - [`f in I: RngMPolLocElt, RngMPolLoc → BoolElt`](ideal-operation.md#operation-op-in-rngmpollocelt-rngmpolloc)

    - [`NormalForm(f, I): RngMPolLocElt, RngMPolLoc → RngMPolLocElt`](ideal-operation.md#function-normalform-rngmpollocelt-rngmpolloc)

    - [`f notin I: RngMPolLocElt, RngMPolLoc → BoolElt`](ideal-operation.md#operation-op-notin-rngmpollocelt-rngmpolloc)

    - [`Example: Element Operations`](ideal-operation.md#example-ex-e2cb48)

- [Changing Coefficient Ring](change-ring.md)

  - [`ChangeRing(I, L): RngMPolLoc, Rng → RngMPolLoc`](change-ring.md#function-rngmpolloc-changering)

- [Changing Monomial Order](change-order.md)

  - [`ChangeOrder(I, Q): RngMPolLoc, RngMPolLoc → RngMPolLoc, Map`](change-order.md#function-rngmpolloc-changeorder)

  - [`ChangeOrder(I, order): RngMPolLoc, ... → RngMPolLoc, Map`](change-order.md#function-changeorder-rngmpolloc)

- [Dimension of Ideals](dimension.md)

  - [`Dimension(I): RngMPolLoc → RngIntElt, [ RngIntElt ]`](dimension.md#function-rngmpolloc-dimension)