Let $\mathcal{P}_{2,Z}$ be the set of all 2 variables quadratic equations $P(x,y)$ with integral coefficients: $$P(x,y)=a_1x^2+a_2y^2+a_3xy+a_4x+a_5y+a_6\ \ \ \ \ \ (a_i\in \mathbb{Z})$$ Consider a system of of $n$ equations: $$\begin{cases} P_1(x_1,x_2)=0\\\ P_2(x_2,x_3)=0\\\ ...\\\ P_n(x_n,x_1)=0\\\ \end{cases} \ \ \ \ \ \\ \ \ \ \ \ \ P_i(x_i,x_{i+1})\in \mathcal{P}_{2,Z};\ \ P_i(x_i,x_{i+1}) \text{ has real roots}.$$ Can we have some constrains of coefficients of $P_i$ such that the system has no real root?
Do you know some work or related materials about this problem? Thank you so much.
$\mathcal{P}_{2,Z}^n$
parameterizing systems with no real root? Of course$U$
contains$V^n$
where$V$
is the open subset of$\mathcal{P}_{2,Z}$
parameterizing conics with no real points. I believe that $V$ is the open subset where each of$a_1D$
,$a_2D$
and$a_6D$
are positive, where $D$ is the discriminant:$D=4a_1a_2a_6+a_3a_4a_5-(a_1a_5^2 + a_2a_4^2+a_6a_3^2)$
. $\endgroup$