Consider the polynomial ring $k[x_0,x_1]$, and the two ideals $I=(x_0,x^2_0 x^2_1,x^3_1)$ and $J=(x^2_0,x^2_1)$. What is the intersection of these ideals?
I found that $I \cap J = (x^2_0,x^2_0x^2_1,x^3_1)$. Is this correct? In general, how can one compute the intersection of two ideals?
you can compute the intersection of two ideals using the following. Introduce a new parameter, say 't'. Find the set $tI + (1-t)J$ and then compute a Groebner basis for this set. so you would have: $$(tx_0,tx_1^2,tx_0^2,tx_1^3,(1-t)x_1^2,(1-t)x_0^2)$$ You can then compute a Groebner basis with respect to a lexicographic order with $t>x_0>x_1$ using Buchberger's Algorithm or autoreduction, whatever you feel more comfortable with. Finally remove all the terms from the Groebner Basis that contain a term with a t in. Then you will have your desired intersection.
You actually have to intersect two monomial ideals, and this is well known; see e.g. here.
In your case we get $I\cap J=(x_0^2,x_0x_1^2,x_1^3)$.
