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^nin 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_{0leq j leq n-1}|a_j|right)^frac{1}{n}right } end{align*} or begin{align*} 1leq |lambda|leq frac{1}{|a_n|}max_{0leq 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].