If I have a curve given by $[X_0^3,X_0X_1^2,X_1^3]\in\mathbb{P}^2$ where $[X_0,X_1]\in \mathbb{P}^1$, then I find that the curve is the zero locus of the polynomial $Z_0Z_2^2-Z_1^3$. How can I find this polynomial using computer software, such as Sage or Singular?
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R=QQ[s,t,x,y,z,MonomialOrder=>Eliminate 2]I=ideal(x-s^3,y-s*t^2,z-t^3)gens gb I -- matrix {{y^3-x*z^2, t*y-s*z, s*y^2-t*x*z, t^2*x-s^2*y, t^3-z, s*t^2-y, s^3-x, s^2*t*z-y^2}}– Jan-Magnus Økland May 31 '22 at 07:19