I need to prove that the ideal $$ I = (xz -y^2, x^2t^2 -yz^3, x^2yt^2 -xz^4) \subset R = \mathbb{K}[x,y,z,t]$$ is generated by a $R$-regular sequence. How can I do it?
I don't know if this can help, but $I = J \cap (x,y)$, where $J$ is generated by the $2\times 2$ minors of $$\begin{pmatrix}x & y & z^3 \\ y & z & xt^2 \end{pmatrix}.$$
In addition, I've calculated the Hilbert series of $R/I$ and I get $$\frac{1 + 2x + 2x^2 + 2x^3 + x^4}{(1-x)^2}$$ so $\dim (R/I) = 2$.
The last generator is superfluous, as
$$(x^2yt^2 - xz^4) - y(x^2t^2 - yz^3) + z^3(xz - y^2) = 0$$
Thus $I = (xz-y^2, x^2t^2-yz^3)$ is generated by a regular sequence (indeed, $xz-y^2, x^2t^2 - yz^3$ are both prime elements).