Non-Archimedean Cauchy Bound for Roots of Non-Archimedean Polynomials

Let $\mathbb{K}$ be a non-Archimedean valued field. Let \begin{align*} p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+a_nz^n\in \mathbb{K}[z], \quad a_n \neq 0. \end{align*} If $\lambda \in \mathbb{K}$ satisfies $p(\lambda)=0$, then we show that \begin{align*} |\lambda|\leq \min \left\{1, \frac{1}{|a_n|^\frac{1}{n}}\left(\max_{0\leq j \leq n-1}|a_j|\right)^\frac{1}{n}\right \} \end{align*} or \begin{align*} 1\leq |\lambda|\leq \frac{1}{|a_n|}\max_{0\leq j \leq n-1}|a_j|. \end{align*} This is the non-Archimedean version of the Cauchy upper bound for every root of a complex polynomial derived by Cauchy in 1829. Our bound is different from the non-Archimedean bound obtained by Nica and Sprague [Am. Math. Mon., 2023].