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# exteriorPower(...,Strategy=>...) -- choose between Bareiss, Cofactor and Dynamic algorithms

## Synopsis

• Usage:
exteriorPower(M, Strategy => s)
• Inputs:
• M,
• s, , either Bareiss, Cofactor, or Dynamic
• Consequences:
• If s is Bareiss, then the Bareiss algorithm is used; if s is Cofactor, then cofactor expansion is used; if s is Dynamic, then a dynamic programming algorithm is used.

## Description

The ring of M determines the default strategy. If the ring is a polynomial ring or a field (as identified by isField) then the Bareiss algorithm is used. If the ring is a quotient ring (which has not been declared a field by toField), then the Cofactor algorithm is used.The Dynamic algorithm implements a variant of cofactor expansion that caches intermediate results. This strategy introduces some memory overhead, but can be faster than Cofactor, especially with sparse matrices of low degree.

## Further information

• Default value: null
• Function: exteriorPower -- exterior power
• Option key: Strategy -- an optional argument

## Caveat

The Bareiss algorithm returns a ring element that may differ from the actual determinant by a zero divisor in the ring. Thus, an incorrect answer may be computed if the ring contains zero divisors.

## Functions with optional argument named Strategy :

• annihilator(...,Strategy=>...) -- see annihilator -- the annihilator ideal
• associatedPrimes(...,Strategy=>...) -- see associatedPrimes -- find associated primes
• basis(...,Strategy=>...) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• canonicalBundle(...,Strategy=>...) (missing documentation)
• mingens(...,Strategy=>...) -- see Complement -- a Strategy option value
• trim(...,Strategy=>...) -- see Complement -- a Strategy option value
• compose(Module,Module,Module,Strategy=>...) -- see compose -- composition as a pairing on Hom-modules
• cotangentSheaf(...,Strategy=>...) (missing documentation)
• determinant(...,Strategy=>...) -- choose between Bareiss, Cofactor and Dynamic algorithms
• dual(ChainComplex,Strategy=>...) (missing documentation)
• dual(Matrix,Strategy=>...) (missing documentation)
• dual(MonomialIdeal,List,Strategy=>...) -- see dual(MonomialIdeal,Strategy=>...)
• dual(MonomialIdeal,RingElement,Strategy=>...) -- see dual(MonomialIdeal,Strategy=>...)
• dual(MonomialIdeal,Strategy=>...)
• dual(SheafMap,Strategy=>...) (missing documentation)
• End(...,Strategy=>...) -- see End -- module of endomorphisms
• exteriorPower(...,Strategy=>...) -- choose between Bareiss, Cofactor and Dynamic 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)
• Hom(...,Strategy=>...) -- see Hom -- module of homomorphisms
• homomorphism'(...,Strategy=>...) -- see homomorphism' -- get the element of Hom from a homomorphism
• 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(Ideal,Ideal,Strategy=>...) -- see intersect(Ideal,Ideal) -- compute an intersection of a sequence of ideals or modules
• intersect(Module,Module,Strategy=>...) -- see intersect(Ideal,Ideal) -- compute an intersection of a sequence 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, Cofactor and Dynamic algorithms
• parallelApply(...,Strategy=>...) -- see parallelApply -- apply a function to each element in parallel
• primaryComponent(...,Strategy=>...) -- see primaryComponent -- find a primary component corresponding to an associated prime
• pushForward(...,Strategy=>...) (missing documentation)
• quotient(...,Strategy=>...)
• 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=>...)
• sheafHom(...,Strategy=>...) (missing documentation)
• primaryDecomposition(...,Strategy=>...) -- see strategies for computing primary decomposition
• syz(...,Strategy=>...) -- see syz(Matrix) -- compute the syzygy matrix
• tangentCone(...,Strategy=>...) -- see tangentCone(Ideal)
• tangentSheaf(...,Strategy=>...) (missing documentation)