Everything in this post is over the complex numbers. I would like to know if for every $\epsilon > 0$ there exists $\delta > 0$ such that the following holds for every $n$ and every $d$ which is sufficiently large depending on $n$.

Let $p_1,\ldots,p_n$ be a set of homogeneous degree two polynomials in the variables $x_1,\ldots,x_d$. 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.

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