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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.

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  • $\begingroup$ Just to clarify: are you asking for a description of the open subset $U$ of $\mathcal{P}_{2,Z}^n$ 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: $D=4a_1a_2a_6+a_3a_4a_5-(a_1a_5^2 + a_2a_4^2+a_6a_3^2)$. $\endgroup$ – Jason Starr Mar 29 '13 at 12:01
  • $\begingroup$ Actually . . . the open set $U$ contains the union from $i=1,\dots,n$ of the inverse image of $V$ under the projection $\pi_i:\mathcal{P}^n_{2,Z}\to \mathcal{P}_{2,Z}$ onto the $i^{\text{th}}$ factor. If any of the equations in the system is inconsistent, then the whole system is inconsistent. $\endgroup$ – Jason Starr Mar 29 '13 at 12:11
  • $\begingroup$ Thank for your suggestion, but in my problem, every P_i has many real roots. I forgot to write that. $\endgroup$ – Kim Jun-Lee Mar 30 '13 at 2:27
  • $\begingroup$ Could you please clarify your question. What precisely are you asking? $\endgroup$ – Jason Starr Mar 30 '13 at 11:45

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