Let $Q_s(x),\hat{Q}_{\hat s}(x)\in \Bbb Z_B[x]$ be pair of homogeneous quadratic forms with $x\in \Bbb C^s$ and $x\in \Bbb C^\hat{s}$ respectively and having coefficients bounded by $B\in\Bbb Z_+$.

Let $S_p(r)$ and $\hat{S}_\hat{p}(\hat{r})$ be spheres of radii $r$ and $\hat r$ centered at $p \in \Bbb C^s$ and at $\hat p \in \Bbb C^\hat{s}$ respectively.

Let $Z_Q$, $Z_{\hat{Q}}$, $Z_S$ and $Z_\hat S$ be zero sets of $Q_s(x)$, $\hat{Q}_{\hat s}(x)$, $S_p(r)$ and $\hat S_\hat p(\hat r)$ respectively.

Let $|x|_1$ be either sum of coordinates function and let $|x|_2$ be sum of squares of coordinates function.

We call $[Q_s(x),\hat{Q}_\hat s(x)]_{(p,r,\hat p,\hat r,i,t)}$ a $(p,r,\hat p,\hat r,i,t)$-pair if $x \in Z_S\cap Z_Q\iff y \in Z_\hat S\cap Z_\hat{Q}$ such that $|x|_i+|y|_i=t \leq s+\hat s$ for a fixed $t$ for either $i=1$ or $2$.

Given a homogeneous $Q_s(x)$, is there always a homogeneous $\hat{Q}_\hat s(x)$? If there is a pair, how do one find a homogeneous $\hat{Q}_\hat s=(x)$ that is a pair? Does the minimum $\hat{s}$ required grow atmost as fast a $O(n^c)$ for some $c>0$?