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:`$D=4a_1a_2a_6+a_3a_4a_5-(a_1a_5^2 + a_2a_4^2+a_6a_3^2)$`

. – Jason Starr Mar 29 '13 at 12:01anyof the equations in the system is inconsistent, then the whole system is inconsistent. – Jason Starr Mar 29 '13 at 12:11preciselyare you asking? – Jason Starr Mar 30 '13 at 11:45