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Let $Q_s(x)\in\Bbb Z[x_1,x_2,\dots,x_s],\hat{Q}_{\hat s}(y)\in \Bbb Z[y_1,y_2,\dots,y_\hat s]$ be pair of homogeneous purely non-diagonal (every term of form $x_ix_j$ or $y_iy_j$) quadratic forms and having coefficients bounded by functions $f(s),\hat{f}(\hat s)$ 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 $S_{r}(x-p)=0$ and $\hat{S}_{\hat{r}}(y-\hat{p})=0$ 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 $Q_s(x)$, $S_{r}(x-p)$ respectively and $Z_{\hat{Q}}$,$Z_\hat S$ be zero sets in $\Bbb F^\hat{s}$ of $\hat{Q}_{\hat s}(y)$ and $\hat{S}_{\hat{r}}(y-\hat{p})$ respectively.

Let $G_1(x)$ be either sum of coordinates function and let $G_2(x)$ be sum of squares of coordinates function.

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

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

Let $Q_s(x)\in\Bbb Z[x_1,x_2,\dots,x_s],\hat{Q}_{\hat s}(y)\in \Bbb Z[y_1,y_2,\dots,y_\hat s]$ be pair of homogeneous purely non-diagonal (every term of form $x_ix_j$ or $y_iy_j$) quadratic forms and having coefficients bounded by functions $f(s),\hat{f}(\hat s)$ 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 $\Bbb F=\Bbb C$ or $\Bbb R$.

Let $S_{r}(x-p)=0$ and $\hat{S}_{\hat{r}}(y-\hat{p})=0$ 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 $Q_s(x)$, $S_{r}(x-p)$ respectively and $Z_{\hat{Q}}$,$Z_\hat S$ be zero sets in $\Bbb F^\hat{s}$ of $\hat{Q}_{\hat s}(y)$ and $\hat{S}_{\hat{r}}(y-\hat{p})$ respectively.

Let $G_1(x)$ be either sum of coordinates function and let $G_2(x)$ be sum of squares of coordinates function.

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

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

Let $Q_s(x)\in\Bbb Z[x_1,x_2,\dots,x_s],\hat{Q}_{\hat s}(y)\in \Bbb Z[y_1,y_2,\dots,y_\hat s]$ be pair of homogeneous purely non-diagonal (every term of form $x_ix_j$ or $y_iy_j$) quadratic forms and having coefficients bounded by functions $f(s),\hat{f}(\hat s)$ 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 $S_{r}(x-p)=0$ and $\hat{S}_{\hat{r}}(y-\hat{p})=0$ 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 $Q_s(x)$, $S_{r}(x-p)$ respectively and $Z_{\hat{Q}}$,$Z_\hat S$ be zero sets in $\Bbb F^\hat{s}$ of $\hat{Q}_{\hat s}(y)$ and $\hat{S}_{\hat{r}}(y-\hat{p})$ respectively.

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

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

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

Let $Q_s(x)\in\Bbb Z[x_1,x_2,\dots,x_s],\hat{Q}_{\hat s}(y)\in \Bbb Z[y_1,y_2,\dots,y_\hat s]$ be pair of homogeneous purely non-diagonal (every term of form $x_ix_j$ or $y_iy_j$) quadratic forms and having coefficients bounded by functions $f(s),\hat{f}(\hat s)$ 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 $S_{r}(x-p)=0$ and $\hat{S}_{\hat{r}}(y-\hat{p})=0$ 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 $Q_s(x)$, $S_{r}(x-p)$ respectively and $Z_{\hat{Q}}$,$Z_\hat S$ be zero sets in $\Bbb F^\hat{s}$ of $\hat{Q}_{\hat s}(y)$ and $\hat{S}_{\hat{r}}(y-\hat{p})$ respectively.

Let $G_1(x)$ be either sum of coordinates function and let $G_2(x)$ be sum of squares of coordinates function.

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

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

Let $Q_s(x)\in\Bbb Z[x_1,x_2,\dots,x_s],\hat{Q}_{\hat s}(y)\in \Bbb Z[y_1,y_2,\dots,y_\hat s]$ be pair of homogeneous purely non-diagonal (every term of form $x_ix_j$ or $y_iy_j$) quadratic forms and having coefficients bounded by functions $f(s),\hat{f}(\hat s)$ 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 $S_{r}(x-p)=0$ and $\hat{S}_{\hat{r}}(y-\hat{p})=0$ 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 $Q_s(x)$, $S_{r}(x-p)$ respectively and $Z_{\hat{Q}}$,$Z_\hat S$ be zero sets in $\Bbb F^\hat{s}$ of $\hat{Q}_{\hat s}(y)$ and $\hat{S}_{\hat{r}}(y-\hat{p})$ respectively.

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

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

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

Let $Q_s(x)\in\Bbb Z[x_1,x_2,\dots,x_s],\hat{Q}_{\hat s}(y)\in \Bbb Z[y_1,y_2,\dots,y_\hat s]$ be pair of homogeneous purely non-diagonal (every term of form $x_ix_j$ or $y_iy_j$) quadratic forms and having coefficients bounded by functions $f(s),\hat{f}(\hat s)$ 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 $S_{r}(x-p)=0$ and $\hat{S}_{\hat{r}}(y-\hat{p})=0$ 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 $Q_s(x)$, $S_{r}(x-p)$ respectively and $Z_{\hat{Q}}$,$Z_\hat S$ be zero sets in $\Bbb F^\hat{s}$ of $\hat{Q}_{\hat s}(y)$ and $\hat{S}_{\hat{r}}(y-\hat{p})$ respectively.

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

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

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