Question 1: In A Singular Introduction to Commutative Algebra, page 222, there is written:
How can I check that this isomorphism actually holds? I would really prefer a computational proof (using a computer), because I just can't do this manually for every single case.
I've tried the surjective map $f\!:K[x,y]\!\rightarrow\!K[-t^3+6t,t^4-6t^2]$, defined by $x\mapsto -t^3+6t$ and $y\mapsto t^4-6t^2$, but the SINGULAR code
ring A=0,(x,y),dp; ring B=0,(t),dp; map F=A,-t3+6t,t4-6t2; ideal I=0; setring A; preimage(B,F,I);
tells me that its kernel is $\langle x^4-84x^2y-y^3-32400x^2-360y^2-32400y\rangle$ and not $\langle x^4+6x^2y-y^3\rangle$, so this isn't the right map. Am I supposed to guess what the right map is?
Question 2: If I have $R:=K[x,y,z]/\langle z^2-xy\rangle$, how can I find a suitable subring of $K[t_1,\ldots,t_n]$ (for some $n$), that is isomorphic to $R$?
EDIT: I would just like to add that after restarting the computer and running the code again, SINGULAR, for some mysterious reason, now returned the correct answer $\langle x^4+6x^2y-y^3\rangle$.