Totally $T$-adic functions of small height

Let $\mathbb{F}_q(T)$ be the field of rational functions in one variable over a finite field. We introduce notion totally $T$-adic function: that is algebraic and whose minimal polynomial splits completely completion $\mathbb{F}_q(\!(T)\!)$. give two proofs height nonconstant function bounded away from zero, each which provides sharp lower bound. spend majority paper providing explicit constructions small (via arithmetic dynamics) minimum geometry computer search). also execute large search proves certain kinds $\mathbb{F}_2(T)$ do not exist. The problem whether there exist infinitely many positive remains open. Finally, we consider analogues these notions under additional integrality hypotheses.