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Linear Independence in Quotients of Polynomial Rings

asked 2022-12-17 18:14:06 +0200

Thrash gravatar image

updated 2023-05-19 22:06:49 +0200

FrédéricC gravatar image

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 2022-12-18 03:18:38 +0200

Max Alekseyev gravatar image

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 gravatar imageThrash ( 2022-12-18 13:15:10 +0200 )edit

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Asked: 2022-12-17 18:14:06 +0200

Seen: 148 times

Last updated: Dec 18 '22