The order types of the supports of $p$-adic algebraic Hahn series

Let $p$ be a prime number. In 1986, David Lampert asked if one expands a ${\mathbf{Q}}_p$-algebraic number as a $p$-adic Hahn series, then what is the possible order type of its support? In 2001, Kedlaya made a vague conjecture that the possible ordinals can only be finite ordinals, $\omega$ and ${\omega }^{\omega }$, where $\omega$ is the first infinite ordinal. In this article, we solve Lampert’s problem by confirming Kedlaya’s intuition. Moreover, we completely characterize the ${\mathbf{Q}}_p$-algebraic numbers with order type of support at most $\omega$. As an application, we determine the bi-algebraic set of the $p$-adic shadow map.