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Given a polynomial  f  defined over a complete local field, we construct a biholomorphic change of variables defined in a neighbourhood of infinity which transforms the action  z ↦  f ( z ) to the multiplicative action  z ↦  z  deg( f ) . The relation between this construction and the Böttcher coordinate in complex polynomial dynamics is similar to the relation between the complex uniformization of elliptic curves and Tate's  p ‐adic uniformization. Specifically, this biholomorphism is Galois equivariant, reducing certain questions about the Galois theory of preimages by  f  to questions about multiplicative Kummer theory. As a consequence, we obtain some corollaries regarding Galois irreducibility of preimages of certain points under certain polynomials, as well as the rationality of preimages in one‐parameter families.

Arboreal Galois representations and uniformization of polynomial dynamics