Define a polynomial subalgebra generated by given polynomials

asked 2022-03-28 12:48:28 +0100

Niccon gravatar image

Given a polynomial algebra in some variables, say $\mathbb{Q}[x_1,x_2,x_3,x_4]$, I'd like to define certain subalgebras that are generated by elements, e.g. $x_4$, $x_3^2-x_2x_4$. So I'd like to define $\mathbb{Q}[x_4,x_3^2-x_2x_4]$. As this uses the variables of the bigger space, I assume it needs to be defined as a subalgebra. But I could not seem to find a way of doing so.

I'm actually just interested in the dimensions of the degree-graded spaces of the subalgebra if that makes it easier.

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Welcome to Ask Sage! Thank you for your question.

slelievre gravatar imageslelievre ( 2022-03-28 13:00:59 +0100 )edit

SageMath doesn't have subalgebra functionality conveniently exposed, but you can access some functionality from Singular. Related: Compute minimal number of generators of subring.

rburing gravatar imagerburing ( 2022-03-28 21:52:58 +0100 )edit