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I come up with a family of systems of multivariate quadratic polynomial equations:

Let $n$ be any integer divisible by and greater than 8. Do there

exist complex numbers $z_1,z_2,\ldots,z_{n-1}$ with modules $1$

satisfying the following quadratic equations, $\sum\limits_{r=1}^{2t-1}(-1)^rz_rz_{2t-r}+\sum\limits_{r=2t+1}^{n- 1}(-1)^rz_rz_{n+2t-r}=\frac{2z_{2t}}{(n^2+n+1)^{\frac{1} {2}}},t=1,2,\ldots,\frac{n}{2}-1$, $z_tz_{\frac{n}{2}+t}=z_{2t},t=1,2,\ldots,\frac{n}{2}-1$, and $z_tz_{n-t}=(-1)^t,t=1,2,\ldots,\frac{n}{2}-1$.

Is there any method which may deal with general $n$ in this problem?

share|improve this question
    
A Gröbner basis method ought to work here methinks... but there might be cleverer ways. –  J. M. Sep 17 '11 at 12:44
    
The Groebner basis algorithm works for a given value of $n$, but I don't know how it may work for general $n$. –  Binzhou Xia Sep 18 '11 at 3:14

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