Everything in this post is over the complex numbers. I would like to know if for every $ \epsilon > 0$ there exists $ \delta > 0$ with the following property.
Let $ p_1,\ldots,p_n$ be a set of homogeneous degree two polynomials with in the variables $ x_1,\ldots,x_d$ , where $ d$ is much larger than $ n$ . Assume that all coefficients in the $ p_k$ are $ 1$ .
Consider the systems
\begin{equation*} \mathcal{S}_1 = \begin{cases} p_1(x_1,\ldots,x_d) = a_1 \ p_2(x_1,\ldots,x_d) = a_2 \ \hspace{0.5in} \vdots \ p_n(x_1,\ldots,x_d) = a_n \end{cases} \end{equation*}
and
\begin{equation*} \mathcal{S}_2 = \begin{cases} p_1(x_1,\ldots,x_d) = b_1 \ p_2(x_1,\ldots,x_d) = b_2 \ \hspace{0.5in} \vdots \ p_n(x_1,\ldots,x_d) = b_n \end{cases} \end{equation*}
where $ a_k$ and $ b_k$ have magnitude bounded by one. Suppose there exist solutions $ \overline{\alpha} = (\alpha_1,\ldots,\alpha_d)$ to $ \mathcal{S}_1$ and $ \overline{\beta} = (\beta_1,\ldots,\beta_d)$ to $ \mathcal{S}_2$ , both in the unit ball, such that $ ||\overline{\alpha} – \overline{\beta}||_2 \leq \delta$ . Then for every solution $ \overline{\alpha}’$ to $ \mathcal{S}_1$ in the unit ball there exists a solution $ \overline{\beta}’$ to $ \mathcal{S}_2$ such that $ ||\overline{\alpha}’ – \overline{\beta}’||_2 \leq \epsilon$ .
My intuition is that since the left sides of $ \mathcal{S}_1$ and $ \mathcal{S}_2$ are the same, the varieties they define should be in some sense parallel, as happens in the linear case.
