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# MinimalGenerators -- whether to compute minimal generators and return a trimmed set of generators

## Description

The following returns two minimal generators (Serre's Theorem: a codim 2 Gorenstein ideal is a complete intersection.)

 i1 : S = ZZ/101[a,b] o1 = S o1 : PolynomialRing i2 : i = ideal(a^4,b^4) 4 4 o2 = ideal (a , b ) o2 : Ideal of S i3 : quotient(i, a^3+b^3) 3 3 o3 = ideal (a*b, a - b ) o3 : Ideal of S

Without trimming we would get 4 generators instead.

 i4 : quotient(i, a^3+b^3, MinimalGenerators => false) 3 3 o4 = ideal (a*b, a - b ) o4 : Ideal of S

Sometimes the extra time to find the minimal generators is too large. This allows one to bypass this part of the computation.

 i5 : needsPackage "Truncations" o5 = Truncations o5 : Package i6 : R = ZZ/101[x_0..x_4] o6 = R o6 : PolynomialRing i7 : I = truncate(8, monomialCurveIdeal(R,{1,4,5,9})); o7 : Ideal of R i8 : time gens gb I; -- used 0.0354604s (cpu); 0.0353149s (thread); 0s (gc) 1 428 o8 : Matrix R <-- R i9 : time J1 = saturate(I); -- used 0.434439s (cpu); 0.222265s (thread); 0s (gc) o9 : Ideal of R i10 : time J = saturate(I, MinimalGenerators => false); -- used 0.00014468s (cpu); 0.00014469s (thread); 0s (gc) o10 : Ideal of R i11 : numgens J o11 = 7 i12 : numgens J1 o12 = 7

## Functions with optional argument named MinimalGenerators :

• adjoint(...,MinimalGenerators=>...) (missing documentation)
• associatedPrimes(...,MinimalGenerators=>...) -- see associatedPrimes -- find associated primes
• canonicalBundle(...,MinimalGenerators=>...) (missing documentation)
• compose(Module,Module,Module,MinimalGenerators=>...) -- see compose -- composition as a pairing on Hom-modules
• cotangentSheaf(...,MinimalGenerators=>...) -- see cotangentSheaf(ProjectiveVariety) -- cotangent sheaf of a projective variety
• dual(ChainComplex,MinimalGenerators=>...) (missing documentation)
• dual(Matrix,MinimalGenerators=>...) (missing documentation)
• dual(SheafMap,MinimalGenerators=>...) (missing documentation)
• End(...,MinimalGenerators=>...) -- see End -- module of endomorphisms
• Hom(...,MinimalGenerators=>...) -- see Hom -- module of homomorphisms
• homomorphism'(...,MinimalGenerators=>...) -- see homomorphism' -- get the element of Hom from a homomorphism
• intersect(Ideal,Ideal,MinimalGenerators=>...) -- see intersect(Ideal,Ideal) -- compute an intersection of a sequence of ideals or modules
• intersect(Module,Module,MinimalGenerators=>...) -- see intersect(Ideal,Ideal) -- compute an intersection of a sequence of ideals or modules
• intersectInP(...,MinimalGenerators=>...) -- see intersectInP(...,BasisElementLimit=>...) -- Option for intersectInP
• decompose(Ideal,MinimalGenerators=>...) -- see minimalPrimes -- minimal primes of an ideal
• minimalPrimes(...,MinimalGenerators=>...) -- see minimalPrimes -- minimal primes of an ideal
• primaryDecomposition(...,MinimalGenerators=>...) -- see primaryDecomposition -- irredundant primary decomposition of an ideal
• quotient(...,MinimalGenerators=>...) -- see quotient(Module,Module) -- ideal or submodule quotient
• analyticSpread(...,MinimalGenerators=>...) -- see reesIdeal(...,MinimalGenerators=>...) -- Whether the saturation step returns minimal generators
• distinguished(...,MinimalGenerators=>...) -- see reesIdeal(...,MinimalGenerators=>...) -- Whether the saturation step returns minimal generators
• isLinearType(...,MinimalGenerators=>...) -- see reesIdeal(...,MinimalGenerators=>...) -- Whether the saturation step returns minimal generators
• isReduction(...,MinimalGenerators=>...) -- see reesIdeal(...,MinimalGenerators=>...) -- Whether the saturation step returns minimal generators
• minimalReduction(...,MinimalGenerators=>...) -- see reesIdeal(...,MinimalGenerators=>...) -- Whether the saturation step returns minimal generators
• multiplicity(...,MinimalGenerators=>...) -- see reesIdeal(...,MinimalGenerators=>...) -- Whether the saturation step returns minimal generators
• normalCone(...,MinimalGenerators=>...) -- see reesIdeal(...,MinimalGenerators=>...) -- Whether the saturation step returns minimal generators
• reesAlgebra(...,MinimalGenerators=>...) -- see reesIdeal(...,MinimalGenerators=>...) -- Whether the saturation step returns minimal generators
• reesIdeal(...,MinimalGenerators=>...) -- Whether the saturation step returns minimal generators
• specialFiber(...,MinimalGenerators=>...) -- see reesIdeal(...,MinimalGenerators=>...) -- Whether the saturation step returns minimal generators
• specialFiberIdeal(...,MinimalGenerators=>...) -- see reesIdeal(...,MinimalGenerators=>...) -- Whether the saturation step returns minimal generators
• saturate(...,MinimalGenerators=>...) -- see saturate -- saturation of ideal or submodule
• sheafExt(ZZ,CoherentSheaf,CoherentSheaf,MinimalGenerators=>...) (missing documentation)
• sheafExt(ZZ,CoherentSheaf,SheafOfRings,MinimalGenerators=>...) (missing documentation)
• sheafExt(ZZ,SheafOfRings,CoherentSheaf,MinimalGenerators=>...) (missing documentation)
• sheafExt(ZZ,SheafOfRings,SheafOfRings,MinimalGenerators=>...) (missing documentation)
• sheafHom(...,MinimalGenerators=>...) (missing documentation)
• tangentSheaf(...,MinimalGenerators=>...) -- see tangentSheaf(ProjectiveVariety) -- tangent sheaf of a projective variety
• truncate(InfiniteNumber,Thing,MinimalGenerators=>...) (missing documentation)
• truncate(List,Ideal,MinimalGenerators=>...) (missing documentation)
• truncate(List,Matrix,MinimalGenerators=>...) -- see truncate(List,Matrix) -- truncation of a map of free modules
• truncate(List,Module,MinimalGenerators=>...) -- see truncate(List,Module) -- truncation of the graded ring, ideal or module at a specified degree or set of degrees
• truncate(List,Ring,MinimalGenerators=>...) (missing documentation)
• truncate(ZZ,Ideal,MinimalGenerators=>...) (missing documentation)
• truncate(ZZ,Matrix,MinimalGenerators=>...) (missing documentation)
• truncate(ZZ,Module,MinimalGenerators=>...) (missing documentation)
• truncate(ZZ,Ring,MinimalGenerators=>...) (missing documentation)

## For the programmer

The object MinimalGenerators is .