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How to compute top-degree homogenization of ideal

asked 2024-09-23 00:52:37 +0200

Atropos7 gravatar image

updated 2024-09-23 16:20:55 +0200

Max Alekseyev gravatar image

I am interested in computing the "top-degree homogenization of an ideal", the ideal generated by the top-degree homogeneous components of every element in a polynomial ideal $I$.

I had previously been (mistakenly) doing:

ideal([f(list(R.gens()) + [0]) for f in I.homogenize().gens()])

which gives the ideal generated by the top-degree homogeneous components of the given generators of $I$.

This is not the same: consider $I = (x^2 + x, x^2 + y)$. The above code would produce $(x^2)$, but the top-degree homogenization should contain $x-y$ (as the top-degree component of $(x^2+x) - (x^2 + y)$).

Are there known techniques for doing this?

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answered 2024-09-23 16:19:39 +0200

Max Alekseyev gravatar image

updated 2024-09-23 16:19:48 +0200

This should do the job:

R.<x,y> = QQ[]
I = R.ideal([x^2+x, x^2+y])
J = R.ideal([f.homogeneous_components()[f.degree()] for f in I.groebner_basis()])
print(J)
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Thank you! It had totally slipped my mind that the top-degree components of a groebner basis for $I$ would be a groebner basis for the ideal I want.

Atropos7 gravatar imageAtropos7 ( 2024-09-23 18:51:05 +0200 )edit

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Asked: 2024-09-23 00:52:37 +0200

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Last updated: Sep 23