Let $Q_s(x)\in\Bbb Z_{f(s)}[x],\hat{Q}_{\hat s}(x)\in \Bbb Z_{\hat{f}(\hat s)}[x]$ be pair of homogeneous purely non-diagonal(every term of form $x_ix_j$) quadratic forms and having coefficients bounded by functions $f(s),\hat{f}(\hat s)\in\Bbb Z_+$ such that $f(s)+\hat{f}(\hat s) \leq (s+\hat{s})B$ for a fixed $B\in\Bbb Z_{\geq 2}$ .

Fix a char $0$ ground field $\Bbb F$.

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

Let $Z_Q$, $Z_S$ be zero sets in $\Bbb F^s$ of $Q_s(x)$, $S_p(r)$ respectively and $Z_{\hat{Q}}$,$Z_\hat S$ be zero sets in $\Bbb F^\hat{s}$ of $\hat{Q}_{\hat s}(x)$ 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 $[Q_s(x),\hat{Q}_\hat s(x)]_{(Z_S,Z_\hat S,i,t)}$ a $(Z_S,Z_\hat S,i,t)$-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 does one find a homogeneous $\hat{Q}_\hat s(x)=0$ that is a pair? Does the minimum $\hat{s}$ required grow atmost as fast a $O(n^c)$ for some $c>0$?