Compute equalizer of maps between polynomial rings?

[John H Palmieri]

I have a polynomial ring R = k[x1, x2, ..., xn] and a ring homomorphism f: R -> R. In case it matters, k=GF(2). I would like to find the subring of elements x satisfying f(x) = x: that is, I want to find the equalizer of the pair of maps (f, 1). Is there anything in Sage that will compute this? The more polynomial generators this can handle, the better.

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John

[Dima Pasechnik]

On Thu, 25 Aug 2022, 00:38 John H Palmieri, <jhpalm...@gmail.com> wrote:

I have a polynomial ring R = k[x1, x2, ..., xn] and a ring homomorphism f: R -> R. In case it matters, k=GF(2). I would like to find the subring of elements x satisfying f(x) = x: that is, I want to find the equalizer of the pair of maps (f, 1). Is there anything in Sage that will compute this? The more polynomial generators this can handle, the better.

Is this subring finitely generated? Invariant theory in positive characteristic is full of surprises...

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John

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[pedrito...@gmail.com]

Dear John,

Wouldn’t be of some help to consider the kernel of  f-Id (with Id the identity map)?

Best,

Pedro

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[John H Palmieri]

Good question, and I don't whether the subring is finitely generated. I want to compute examples — what's the subring in a range of degrees — to see what's going on.

[John H Palmieri]

One issue is that f-id is not a ring homomorphism. So do I restrict to a range of degrees, convert to vector spaces, and compute the kernel? I'm not sure of the right approach.

[pedrito...@gmail.com]

Yes, probably working up to some degree. I do no know if this could help.

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