of the form ${\mathbf{U}}=\prod_{\nu\in S} {\mathbf{U}}_{\nu}  $ where ${\mathbf{U}}_\nu\subset{\mathbb{Q}}_\nu^{d_\nu}  $ is an open box. Here, the norm is taken to be the Euclidean norm at the infinite place and the $L^{\infty}$ norm at finite places.

3.  We will consider functions ${\mathbf{f}}({\mathbf{x}})  =({\mathbf{f}}_\nu(x_\nu)) _{\nu\in S}$, for ${\mathbf{x}}=(x_\nu) \in{\mathbf{U}}$ where ${\mathbf{f}}_\nu=(f_\nu^{(1)},f_\nu^{(2)},\dots,f_\nu^{(n)}): 
                      {\mathbf{U}}_\nu\to {\mathbb{Q}}_\nu^n $ is an analytic map for any $\nu\in S $, and can be analytically extended to the boundary of $ {\mathbf{U}}_\nu$.

4.  We assume that the restrictions of $1 ,{f_\nu^{(1)},f_\nu^{(2)},\dots,f_\nu^{(n)}}$ to any open subset of ${\mathbf{U}}_\nu $ are linearly independent over ${\mathbb{Q}}_\nu $ and that $\|{\mathbf{f}}({\mathbf{x}})\|\leq 1,\|\nabla{\mathbf{f}}_\nu(x_\nu)\| \leq 1$ and $|\Phi_\beta {\mathbf{f}}_\nu(y_1,y_2,y_3)| \leq \frac{1}{2} $ for any $\nu \in S,$ second difference quotient $\Phi_\beta$ and $x_\nu,y_1,y_2,y_3 \in U_\nu$. We refer the reader to Section $3$ for definitions.

5.  \[monotone\_cond\] We assume that the function $\Psi :{\mathbb{Z}}^n \to {\mathbb{R}}_{+ } $ is monotone decreasing componentwise i.e. $$\Psi(a_1,\cdots,a_i,\cdots, a_n)\geq \Psi(a_1,\cdots, a'_{i},\cdots, a_n)$$ whenever $|a_i|_S\leq |a'_i|_S $.

6.  We assume that $\Theta({\mathbf{x}})=(\Theta_\nu(x_\nu)) $ where $\Theta :{\mathbf{U}}\mapsto {\mathbb{Q}}_S $ is also analytic and can be extended analytically to the boundary of ${\mathbf{U}}_\nu$.we will assume $\|\Theta({\mathbf{x}})\|\leq 1,\|\nabla\Theta_\nu(x_\nu)\| \leq 1$ and $|\Phi_\beta \Theta_\nu(y_1,y_2,y_3)| \leq \frac{1}{2} $ for any $\nu \in S $ , second difference 