Is there a Grobner basis method that can compute the intersection of an ideal $I$ of a polynomial ring $R$ and its subring $R^\prime$? For example, I have an ideal $I=(x+y+z^2,1+xyz+yz+xz)$ of $\mathbb{F}_2[x,y,z]$ and want to find its intersection with the subring $\mathbb{F}_2[x^2,y^2,z^2]$
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2$\begingroup$ You can apply elimination theory (see e.g. §15.10.4 of Eisenbud) to the ideal $$(x+y+z^2,1+xyz+yz+xz,x^2-a,y^2-b,z^2-c) \subseteq \mathbf F_2[a,b,c,x,y,z].$$ I think in general you're supposed to reduce to such a situation (although I may be mistaken ― I am far from an expert). I guess the point is that if $R'$ is not a polynomial ring, we want to work with a presentation of $R'$ anyway. $\endgroup$– R. van Dobben de BruynCommented May 21, 2019 at 1:12
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1$\begingroup$ mathoverflow.net/questions/324762/… $\endgroup$– Jason StarrCommented May 25, 2019 at 11:46
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