quotient(...,Strategy=>...)

Description

i1 : hooks methods(quotient, Ideal, Ideal)

o1 = {0 => (quotient, Ideal, Ideal, Strategy => Quotient)}
     {1 => (quotient, Ideal, Ideal, Strategy => Iterate) }
     {2 => (quotient, Ideal, Ideal, Strategy => Monomial)}

o1 : NumberedVerticalList

i2 : code(quotient, Ideal, Ideal, Strategy => Quotient)

o2 = -- code for method: quotient(Ideal,Ideal)
     /usr/share/Macaulay2/Saturation.m2:222:30-233:23: --source code:
         Quotient => (opts, I, J) -> (
             R := ring I;
             mR := transpose generators J ** (R / I);
             -- if J is a single element, this is the same as
             -- computing syz gb(matrix{{f}} | generators I, ...)
             g := syz gb(mR, opts,
                 Strategy   => LongPolynomial,
                 Syzygies   => true,
                 SyzygyRows => 1);
             -- The degrees of g are not correct, so we fix that here:
             -- g = map(R^1, null, g);
             lift(ideal g, R)),

i3 : n = 6

o3 = 6

i4 : S = ZZ/101[vars(0..n-1)];

i5 : I = monomialCurveIdeal(S, 1..n-1);

o5 : Ideal of S

i6 : J = ideal(map(S^1, S^n, (p, q) -> S_q^5));

o6 : Ideal of S

i7 : time quotient(I^3, J^2, Strategy => Iterate);
     -- used 0.558153 seconds

o7 : Ideal of S

i8 : time quotient(I^3, J^2, Strategy => Quotient);
     -- used 0.784608 seconds

o8 : Ideal of S

i9 : S = ZZ/101[vars(0..4)];

i10 : I = ideal vars S;

o10 : Ideal of S

i11 : time quotient(I^5, I^3, Strategy => Iterate);
     -- used 0.0546585 seconds

o11 : Ideal of S

i12 : time quotient(I^5, I^3, Strategy => Quotient);
     -- used 0.00829957 seconds

o12 : Ideal of S

Further information

• Default value: null
• Function: quotient -- quotient or division
• Option key: Strategy -- an optional argument

References

Functions with optional argument named Strategy :

• "addHook(...,Strategy=>...)" -- see addHook -- add a hook function to an object for later processing
• "annihilator(...,Strategy=>...)" -- see annihilator -- the annihilator ideal
• "associatedPrimes(...,Strategy=>...)" -- see associatedPrimes -- find associated primes
• "mingens(...,Strategy=>...)" -- see Complement -- a Strategy option value
• "trim(...,Strategy=>...)" -- see Complement -- a Strategy option value
• determinant(...,Strategy=>...) -- choose between Bareiss and Cofactor algorithms
• "dual(MonomialIdeal,List,Strategy=>...)" -- see dual(MonomialIdeal,Strategy=>...)
• "dual(MonomialIdeal,RingElement,Strategy=>...)" -- see dual(MonomialIdeal,Strategy=>...)
• dual(MonomialIdeal,Strategy=>...)
• exteriorPower(...,Strategy=>...) -- choose between Bareiss and Cofactor algorithms
• "gb(...,Strategy=>...)" -- see gb -- compute a Gröbner basis
• gcdLLL(...,Strategy=>...) (missing documentation)
• "GF(...,Strategy=>...)" -- see GF -- make a finite field
• "groebnerBasis(...,Strategy=>...)" -- see groebnerBasis -- Gröbner basis, as a matrix
• hermite(...,Strategy=>...) (missing documentation)
• "hooks(...,Strategy=>...)" -- see hooks -- list hooks attached to a key
• "idealizer(...,Strategy=>...)" -- see idealizer -- compute Hom(I,I) as a quotient ring
• integralClosure(...,Strategy=>...) -- control the algorithm used
• "intersect(...,Strategy=>...)" -- see intersect(List) -- compute an intersection of ideals or modules
• "intersectInP(...,Strategy=>...)" -- see intersectInP(...,BasisElementLimit=>...) -- Option for intersectInP
• "isPrimary(...,Strategy=>...)" -- see isPrimary -- determine whether a submodule is primary
• "isPrime(Ideal,Strategy=>...)" -- see isPrime(Ideal) -- whether an ideal is prime
• LLL(...,Strategy=>...) -- choose among different algorithms
• "localize(...,Strategy=>...)" -- see localize -- localize an ideal at a prime ideal
• "match(...,Strategy=>...)" -- see match -- regular expression matching
• "decompose(Ideal,Strategy=>...)" -- see minimalPrimes -- minimal primes of an ideal
• "minimalPrimes(...,Strategy=>...)" -- see minimalPrimes -- minimal primes of an ideal
• minors(...,Strategy=>...) -- choose between Bareiss and Cofactor algorithms
• "primaryComponent(...,Strategy=>...)" -- see primaryComponent -- find a primary component corresponding to an associated prime
• quotient(...,Strategy=>...)
• "radical(...,Strategy=>...)" -- see radical -- the radical of an ideal
• "radicalContainment(...,Strategy=>...)" -- see radicalContainment -- whether an element is contained in the radical of an ideal
• "analyticSpread(...,Strategy=>...)" -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• "distinguished(...,Strategy=>...)" -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• "isLinearType(...,Strategy=>...)" -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• "isReduction(...,Strategy=>...)" -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• "minimalReduction(...,Strategy=>...)" -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• "multiplicity(...,Strategy=>...)" -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• "normalCone(...,Strategy=>...)" -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• "reesAlgebra(...,Strategy=>...)" -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• "specialFiber(...,Strategy=>...)" -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• "specialFiberIdeal(...,Strategy=>...)" -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• "regSeqInIdeal(...,Strategy=>...)" -- see regSeqInIdeal -- a regular sequence contained in an ideal
• resolution(...,Strategy=>...)
• saturate(...,Strategy=>...)
• "primaryDecomposition(...,Strategy=>...)" -- see strategies for computing primary decomposition
• "syz(...,Strategy=>...)" -- see syz(Matrix) -- compute the syzygy matrix
• "tangentCone(...,Strategy=>...)" -- see tangentCone(Ideal)