5 problems

Polynomial ramification criterion. For every positive integer $b$, there exists a polynomial $P_b\in\mathbb{Z}[x_1,\dots,x_{b+1}]$ such that $f$ is $b$-ramified if and only if…

Let $F$ be a field of characteristic $p$, let $\varphi$ be an analytic map from the open unit polydisk in $n$ dimensions to itself, and let $x=(0,\ldots,0)$ be a fixed point whose…

Let $F$ be a field of positive characteristic, let $\varphi:\mathbb{P}^{1}_{F}\to\mathbb{P}^{1}_{F}$, and let $0$ be a fixed point with multiplier $\lambda$. Isolation conjecture.…

Totally ramified extension conjecture. If $phi$ has potentially good reduction, then there is a totally ramified extension $L/K$ with $[L:K]\leq B$ such that $phi$ is conjugate ove…

Let $K$ be an algebraically closed complete nonarchimedean field, possibly with trivial absolute value, and let $X$ be an irreducible projective variety over $K$. Suppose…