Linear Independence in Quotients of Polynomial Rings

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asked 2 years ago

Thrash

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updated 1 year ago

FrédéricC

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Consider the following:

K = GF(2)    # can be an arbitrary field
R.<a,b> = PolynomialRing(K)
I = R.ideal(a^2-1,a*b)
Q = R.quo(I)

Now $1,a,b$ are linearly dependent in Q because $b=0$ (in Q).

Is there an elegant way to check this with Sage, especially when the ideals get more complicated? That is: Is there a method to check whether $1,a_1,\dots,a_n$ are linearly (in)dependent in a given quotient $K[a_1,\dots,a_n]/I$ ?

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answered 2 years ago

Max Alekseyev

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Use order that depends on degree (eg. degrevlex), compute Grobner basis wrt such order, and check if it contains any polynomials of degree 1. Any such polynomial will gives a desired linear combination:

K = GF(2)    # can be an arbitrary field
R.<a,b> = PolynomialRing(K,order='degrevlex')
I = R.ideal(a^2-1,a*b)
print([f for f in I.groebner_basis() if f.degree()==1])

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Thanks, I've already played with that, too. In this case this should be equivalent to the fact that if we define all the multiplications, that is $a_ia_j - \dots$ (in my example one would have to add something of the form $b^2 - \dots$ to the ideal), then the Gröbner basis should have the length 3 (in my example) if and only if $1,a,b$ are linearly independent in the related quotient. I just wonder if there is a faster method.

Thrash ( 2 years ago )

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