Let $k$ be any field, and we give $k^3$ the Zariski topology.
Then the question is how to compute the Zariski closure of the set $S=\{(x,y,z)\in k^3|xz=y, x+1=z^2, x\neq0\}$ in $k^3$.
The motivation is that I want to compute the blowup of the variety $\{(x,y)\in k^2|x^3+x^2=y^2\}$ at the point $(0,0)$.
Remarks: I have seen this question and this question. I also want more examples of computing the Zariski closure.
Merci beaucoup !