lilmoore11p8

2022-07-16

How can I show that there are only finitely many solutions for the following system?

${x}^{2}+yz=x$

${y}^{2}+zx=y$

${z}^{2}+xy=z$

${x}^{2}+yz=x$

${y}^{2}+zx=y$

${z}^{2}+xy=z$

persstemc1

Beginner2022-07-17Added 18 answers

First assume $x=y=z$. That will give you two solutions.

Otherwise, one of the three unknowns differs from both others. If we assume $x\ne y$ and $x\ne z$, use lab bhatteacharjee's hint to obtain two linear equations in $x,y,z$. You will notice that $y=z$ and $x=1$ follows from these and then there is a unique solution of the original equations (plus two others, by symmetry)

Otherwise, one of the three unknowns differs from both others. If we assume $x\ne y$ and $x\ne z$, use lab bhatteacharjee's hint to obtain two linear equations in $x,y,z$. You will notice that $y=z$ and $x=1$ follows from these and then there is a unique solution of the original equations (plus two others, by symmetry)

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?

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