Local Polynomial Rings

• Introduction
• Elements and Local Monomial Orders
  • Local Lexicographical: llex
  • Local Graded Lexicographical: lglex
  • Local Graded Reverse Lexicographical: lgrevlex
• Local Polynomial Rings and Ideals
  • Creation of Local Polynomial Rings and Accessing their Monomial Orders
    • LocalPolynomialRing(K, n): Rng, RngIntElt → RngMPolLoc
    • LocalPolynomialRing(K, n, order): Rng, RngIntElt, MonStgElt, ... → RngMPolLoc
    • LocalPolynomialAlgebra(K, n, order): Rng, RngIntElt, MonStgElt, ... → RngMPolLoc
    • LocalPolynomialRing(K, n, T): Rng, RngIntElt, Tup → RngMPolLoc
    • MonomialOrder(R): RngMPolLoc → Tup
    • MonomialOrderWeightVectors(R): RngMPol → [ [ FldRatElt ] ]
    • Localization(R): RngMPol → RngMPolLoc
    • Localization(I): RngMPol → RngMPolLoc
    • Example: Order
  • Creation of Ideals and Accessing their Bases
    • ideal<R | L>: RngMPolLoc, List → RngMPolLoc
    • Ideal(B): [ RngMPolLocElt ] → RngMPolLoc
    • Ideal(B): { RngMPolLocElt } → RngMPolLoc
    • Ideal(f): RngMPolLocElt → RngMPolLoc
    • IdealWithFixedBasis(B): [ RngMPolLocElt ] → RngMPolLoc
    • Basis(I): RngMPolLoc → [ RngMPolLocElt ]
    • BasisElement(I, i): RngMPolLoc, RngIntElt → RngMPolLocElt
• Standard Bases
  • Construction of Standard Bases
    • StandardBasis(I): RngMPolLoc → RngMPolLocElt
    • StandardBasis(S): [ RngMPolLocElt ] → [ RngMPolLocElt ]
    • StandardBasis(S): { RngMPolLocElt } → [ RngMPolLocElt ]
    • Coordinates(I, f): RngMPolLoc, RngMPolLocElt → [ RngMPolLocElt ]
    • CoordinateMatrix(I): RngMPolLoc → Matrix
    • Example: Standard Basis
    • Example: Standard Basis2
• Operations on Ideals
  • Basic Operations
    • I + J: RngMPolLoc, RngMPolLoc → RngMPolLoc
    • I * J: RngMPolLoc, RngMPolLoc → RngMPolLoc
    • I ^ k: RngMPolLoc, RngIntElt → RngMPolLoc
    • QuotientDimension(I): RngMPol → RngIntElt
    • Generic(I): RngMPolLoc → RngMPolLoc
    • LeadingMonomialIdeal(I): RngMPolLoc → RngMPolLoc
    • I meet J: RngMPolLoc, RngMPolLoc → RngMPolLoc
    • &meet S: [ RngMPolLoc ] → RngMPolLoc
  • Ideal Predicates
    • I eq J: RngMPolLoc, RngMPolLoc → BoolElt
    • I ne J: RngMPolLoc, RngMPolLoc → BoolElt
    • I notsubset J: RngMPolLoc, RngMPolLoc → BoolElt
    • I subset J: RngMPolLoc, RngMPolLoc → BoolElt
    • IsZero(I): RngMPolLoc → BoolElt
    • IsProper(I): RngMPolLoc → BoolElt
    • IsZeroDimensional(I): RngMPolLoc → BoolElt
    • Example: Ideal Arithmetic
  • Operations on Elements of Ideals
    • f in I: RngMPolLocElt, RngMPolLoc → BoolElt
    • NormalForm(f, I): RngMPolLocElt, RngMPolLoc → RngMPolLocElt
    • f notin I: RngMPolLocElt, RngMPolLoc → BoolElt
    • Example: Element Operations
• Changing Coefficient Ring
  • ChangeRing(I, L): RngMPolLoc, Rng → RngMPolLoc
• Changing Monomial Order
  • ChangeOrder(I, Q): RngMPolLoc, RngMPolLoc → RngMPolLoc, Map
  • ChangeOrder(I, order): RngMPolLoc, ... → RngMPolLoc, Map
• Dimension of Ideals
  • Dimension(I): RngMPolLoc → RngIntElt, [ RngIntElt ]