The perfectoid Tate algebra has uncountable Krull dimension

Let $K$ be a perfectoid field with pseudo-uniformizer $\pi$. We adapt an argument of Du in \cite{DuUncountable} to show that the perfectoid Tate algebra $K⟨x^{1\mathrm{/}{p}^{\mathrm{\infty }}}⟩$ has an uncountable chain of distinct prime ideals. First, we conceptualize Du’s argument, defining the notion of a $\mathit{\text{Newton}}\mathit{\text{polygon}}\mathit{\text{formalism}}$ on a ring. We prove a version of Du’s theorem in the prescence of a sufficiently nondiscrete Newton polygon formalism. Then, we apply our framework to the perfectoid Tate algebra via a “nonstandard” Newton polygon formalism (roughly, the roles of the series variable $x$ and the pseudo-uniformizer $\pi$ are switched). We conclude a similar statement for multivatiate perfectoid Tate algebras using the one-variable case.