If $n$ is a positive integer and $b_1, b_2, \cdots, b_n$ are positive real numbers such that $b_i \not= b_{i+1}$ (with the convention that $b_{n+1} = b_1$), then can it be shown that the system of equations $a_i^2+a_ia_{i+1}+a_{i+1}^2 = b_i$ for $i=1,2,\cdots, n$ has a finite number of solutions? (EDIT: Gerry Myerson pointed out that I forgot to write $a_{n+1} = a_1.$)
I'm curious about this question because the corresponding questions for $x+y, xy, x^2+y^2, x^2+y^2+2xy$ all can have infinite numbers of solutions, while intuitively, this curve shouldn't because there aren't any obvious reductions to the linear case like in the previous examples.
I labeled this algebraic geometry because it seemed to be a problem that was perhaps solvable with the methods of algebraic geometry (In particular, it seems that if the variables are required to be complex rather than real, the statement should still hold.)
For n odd and all the variables positive reals, it can be shown there is at most 1 solution. If $a_1$ increases, then $a_2$ decreases, and $a_3$ increases, and so on until it can be shown that $a_1$ has both increased and decreased. A similar argument holds for when $a_1$ decreases, so $a_1$ must stay the same.