Linear Independence in Quotients of Polynomial Rings

asked 2022-12-17 18:14:06 +0100

Thrash

515 ●9 ●20 ●38

updated 2023-05-19 22:06:49 +0100

FrédéricC

5251 ●3 ●44 ●116

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 $K[a_1,\dots,a_n]/I$?

edit retag flag offensive close merge delete

add a comment

1 Answer

Sort by » oldest newest most voted

answered 2022-12-18 03:18:38 +0100

Max Alekseyev

6488 ●9 ●47 ●153

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])

edit flag offensive delete link

Comments

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 ( 2022-12-18 13:15:10 +0100 )edit

add a comment

Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

Question Tools

2 followers

Stats

Asked: 2022-12-17 18:14:06 +0100

Seen: 223 times

Last updated: Dec 18 '22

Linear Independence in Quotients of Polynomial Rings edit