Task: the removal of the (1/u) pole in such a way that the c...

Prompt

Task: the removal of the (1/u) pole in such a way that the constant determinant and the collision of the two distinct points are preserved as well We define: [ \boxed{ R_1(u,v)=2u^3v+\frac1u,\qquad R_2(u,v)=u^6v^2+u^2v, \qquad u\ne0. } ] Jacobian matrix: [ J_R= \begin{pmatrix} 6u^2v-u^{-2}&2u^3\\ 6u^5v^2+2uv&2u^6v+u^2 \end{pmatrix}. ] The full expansion of the determinant: [ \begin{aligned} \det J_R &=(6u^2v-u^{-2})(2u^6v+u^2) -(2u^3)(6u^5v^2+2uv)\ &=12u^8v^2+4u^4v-1 -12u^8v^2-4u^4v\ &=\boxed{-1}. \end{aligned} ] Collision certificate: [ \boxed{R(1,0)=R(-1,-1)=(1,0).} ] The polynomiality test: [ R_1=\frac{1+2u^4v}{u}, \qquad \left.(1+2u^4v)\right|_{u=0}=1. ]