I came across the following system of polynomial equations on $X_1,\dots,X_{m-2}$:
$$
\begin{cases}
2X_{2s}+\sum\limits_{t=1}^{2s-1}(-1)^tX_tX_{2s-t}=0,\quad s=1,\dots,\frac{m}{2}-1,\\
X_sX_{m-s}+(-1)^s=0,\quad s=2,\dots,\frac{m}{2},\\
X_1X_{\frac{m}{2}}-2X_{\frac{m}{2}+1}=0,
\end{cases}
$$
where $m\ge10$ is an even integer.
By Groebner basis computation, I verified up to $m=20$ that $1$ is in the ideal generated by the above polynomials in $\mathbb{Q}[X_1,\dots,X_{m-2}]$, whence the system has no solution in $\mathbb{C}^{m-2}$. Also, the number of equations is one more than the number of variables. Thus it is reasonable to conjecture that the system has no solution in $\mathbb{C}^{m-2}$ for any even $m\ge10$, and so the question arises naturally as how to prove this.
Remark: For a given value (not too lagre) of $m$, there are ways such as Groebner basis and various kinds of multipolynomial resultants to show that the system has no solution. But it seems to me that these algorithmic ways give not much insight for general $m$.
A further question: If we have known that the system has no solution in $\mathbb{C}^{m-2}$, we immediately deduce that it has no solution in $\mathbb{F}_p^{m-2}$ for all sufficiently large prime $p$. However, how can I then get a bound $N$ such that the system has no solution in $\mathbb{F}_p^{m-2}$ for prime $p>N$? I guess a bound like $p>m$ holds, but achieving this would be ad hoc to the equations.
Best Answer
For the Further question, See Pascal Koiran's paper (which, I believe, is the last word in the subject, even if almost 20 years old).