0$ such that the Newton polygon of $\det(I-XU_p|\ssc_w(G,\usc))$ lies above the curve
\[
y=A_rx^{1+\frac1{2^n-n-1}}-C_rx.
\]
\end{thm}
As late as 2015, Andreatta, Iovita, and Pilloni said in Section 1.2.2 of~\cite{aip15} that there were not even any conjectures about the shape of the Newton polygon in higher dimensions in the literature.
Our precise theorem is as follows.
\begin{thm}
\label{mybounds}
Fix $\usc$.
\begin{enumerate}
\item \label{mylbd} There are constants $A_1,C>0$ such that for all $w$ such that each $|T_i(w)|>\frac1p$, the Newton polygon of the power series $\det(I-XU_p|\ssc_w(G,\usc))$ lies above the curve
\[
y=\left(A_1x^{1+\frac2{n(n-1)}}-C\right)\min_iv(T_i(w)).
\]
\item \label{myubd} Suppose that $w(a_1,\dotsc,a_{n-1})=\prod_i a_i^{t_i}\chi_i(a_i)$, where $(t_1,\dotsc,t_{n-1})\in(\ints_{\ge0})^{n-1}$ with $t_1\ge\dotsb\ge t_{n-1}$, and each $\chi_i$ is a finite character of conductor $c_i$, satisfying technical restrictions. Let $\chi_{(1)},\dotsc,\chi_{(n-1)}$ be the characters $\chi_1,\dotsc,\chi_{n-1}$ reordered so that $\cond(\chi_{(1)})\le\cond(\chi_{(2)})\le\dotsb\le\cond(\chi_{(n-1)})$, let $c_{(i)}=\cond(\chi_{(i)})$, and let $T_{(i)}=T(\chi_{(i)})$.
Then there is a constant $h$, a polynomial $d_{t_1,\dotsc,t_{n-1}}$ of total degree $\frac{n(n-1)}2$ in the $t_i$s, and a linear function $l(t_1,\dotsc,t_{n-1})$ such that the Newton polygon of $\det(I-XU_p|\ssc_w(G,\usc))$ contains at least
\[
hp^{c_{(1)}+2c_{(2)}+\dotsb+(n-1)c_{(n-1)}-\frac{n(n-1)}2}d_{t_1,\dotsc,t_{n-1}}
\]
segments of slope at most $l(t_1,\dotsc,t_{n-1})$, hence passes below the point
\[
\left(hp^{c_{(1)}+2c_{(2)}+\dotsb+(n-1)c_{(n-1)}-\frac{n(n-1)}2}d_t,hp^{c_{(1)}+2c_{(2)}+\dotsb+(n-1)c_{(n-1)}-\frac{n(n-1)}2}d_tl(t)\right)
\]
\[
=\left(x,A_2\left(v(T_{(1)})^{\frac{2}{n(n-1)}}v(T_{(2)})^{\frac{2\cdot 2}{n(n-1)}}\dotsb v(T_{(n-1)})^{\frac{2\cdot(n-1)}{n(n-1)}}\right)x^{1+\frac2{n(n-1)}}\right)
\]
for $x=hp^{c_{(1)}+2c_{(2)}+\dotsb+(n-1)c_{(n-1)}-\frac{n(n-1)}2}d_t$ and a constant $A_2$. Note that in particular,
\[
v(T_{(1)})^{\frac{2}{n(n-1)}}v(T_{(2)})^{\frac{2\cdot 2}{n(n-1)}}\dotsb v(T_{(n-1)})^{\frac{2\cdot(n-1)}{n(n-1)}}\le\max_i v(T_i).
\]
\end{enumerate}
\end{thm}
We also leverage Theorem~\ref{mybounds} to prove two statements that may be more geometrically satisfying. First, we prove the following alternative version of the upper bound which provides infinitely many upper bound points on the same Newton polygon.
\begin{thm}
\label{unified-ubd}
%There is a constant $A_2$ (depending on $n$, $p$, and $h$) such that for every locally algebraic weight $\chi t$ and radius $r>0$, there is a weight $s$ such that $|T_i(\chi t)-T_i(s)|l}$ with coefficients in $p\ints_p$. Similarly,
\[
p(uz)_{ij}=Z_{j,\sg_{ij}/1}(u^{-1}\bar{N}(\un{z}))=\frac{a_{j,\sg,1_j}+\sum_{\#\tau=j,\tau\neq 1_j}a_{j,\sg,\tau}(u)Z_{j,\tau/1}(\bar{N}(\un{z}))}{a_{j,1_j,1_j}+\sum_{\#\tau=j,\tau\neq 1_j}a_{j,1_j,\tau}(u)Z_{j,\tau/1}(\bar{N}(\un{z}))}
\]
where $Z_{j,\tau/1}(\bar{N}(\un{z}))$ is again a polynomial in the variables $\{z_{kl}\}_{l\le j,k>l}$ with coefficients in $p\ints_p$.
%\[
%e_\sg\cdot\it_j(ux)=e_\sg\cdot\it_j(u)\cdot\it_j(x)=\left(\left(\prod_{k\in\sg}u_{kk}\right)e_\sg+\sum_{\tau>\sg}a_\tau(u)e_\tau\right)\cdot\it_j(x)
%e_\sg\cdot\it_j(xb)=e_\sg\cdot\it_j(x)\cdot\it_j(b)=\left(Z_{j,\sg}(x)e_1\w\dotsb\w e_j+\sum_{\tau>\{1,\dotsc,j\}}a_\tau(x)e_\tau\right)
%\]
\subsection{Notation for subgroups of $\iw_p$}
\label{subgpnotation}
Since we will work with numerous subgroups of $\iw_p$, we will introduce some notation to identify them. If $\un{c}=(c_{ij})\in\ints_{\ge0}^{n\times n}$ is any $n\times n$ matrix of nonnegative integers, we will write
\[
\Gam(\un{c})=\{(x_{ij})\in GL_n(\ints_p)\mid p^{c_{ij}}\mid (x_{ij}-\del_{ij})\text{ for all }i,j\}.
\]
One can compute that $\Gam(\un{c})$ is a group precisely when $c_{ij}\le c_{ik}+c_{kj}$ for all $i,j,k$. Note that this means that if $\Gam(\un{c})$ is a group, then so is $T(\ints_p)\Gam(\un{c})$. If we instead only have half a matrix of nonnegative integers $\un{c}=(c_{ij})_{n\ge i>j\ge1}\in\ints_{\ge0}^{n(n-1)/2}$, we will write
\[
\Gam_1(\un{c})=\{(x_{ij})\in\iw_p\mid v(x_{ij})\ge c_{ij}\forall i>j;v(x_{ii}-1)\ge \min\{c_{ij}|j

*i\}\forall i\}%\subset\Gam_1(p)
\]
\[
\Gam_0(\un{c})=\{(x_{ij})\in\iw_p\mid v(x_{ij})\ge c_{ij}\forall i>j\}=T(\ints_p)\Gam_1(\un{c})\subset\iw_p.
\]
\begin{defn}
We say that $\un{c}=(c_{ij})_{n\ge i>j\ge1}\in\ints_{\ge0}^{n(n-1)/2}$ is \emph{group-shaped} if $c_{ij}\le c_{ik}+c_{kj}$ for all $k$, where we set $c_{ab}$ to be $0$ if $a\le b$.
\end{defn}
Thus $\Gam_1(\un{c})$ and $\Gam_0(\un{c})$ are subgroups whenever $\un{c}$ is group-shaped.
\begin{defn}
We call an $n(n-1)/2$-tuple $\un{c}=(c_{ij})_{n\ge i>j\ge1}\in\ints_{\ge0}^{n(n-1)/2}$ \emph{compatible with} an $n$-tuple $(c_1,\dotsc,c_n)\in\ints_{\ge0}^n$ if $c_i\le \min\{c_{ij}|j i\}\forall i$. Equivalently, if we define $\un{c}'\in\ints_{\ge0}^{n\times n}$ by $c_{ij}'=c_{ij}$ for $i>j$, $c_{ii}'=c_i$, and $c_{ij}'=0$ for $i*