reducing a system of polynomial equations over $\overline{\mathbb{F}_p}$.

Question

I am trying to reduce the system of equations \begin{gather} xw^p-zy^p=1\\ x^pw-z^py=1\\ xz^p=x^pz\\ xw-yz=1.\\ \end{gather} over $\overline{\mathbb{F}_p}$. I want to reduce this system to the single equation $XY^p-X^pY=1$.

I tried and got nowhere. Can someone help?

Answer 1

For example, with $p=2$ (but it works for all primes), there is a solution $$ x=1,\; z=1,\;w=y+1 $$ with arbitrary $y\in \overline{\Bbb F_2}$. This does not imply that $xy^p-x^py-1=0$ because we have $$ xy^2-x^2y-1=y^2+y+1\neq 0. $$