On the unification problem for $\mathrm{GLP}$

We show that the polymodal provability logic $\mathrm{GLP}$, in a language with at least two modalities and one variable, has nullary unification type. More specifically, we show that the formula $[1]p$ does not have maximal unifiers, and exhibit an infinite complete set of unifiers for it. Further, we discuss the algorithmic problem of whether a given formula is unifiable in $\mathrm{GLP}$ and remark that this problem has a positive solution. Finally, we state the arithmetical analogues of the unification and admissibility problems for $\mathrm{GLP}$ and formulate a number of open questions.
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