Let be a complete, algebraically closed nonarchimedean valued field, and let have degree . We show there is a canonical way to assign nonnegative integer weights to points of the Berkovich projective line over in such a way that . When has bad reduction, the set of points with nonzero weight forms a distributed analogue of the unique point which occurs when has potential good reduction. Using this, we characterize the minimal resultant locus of in analytic and moduli-theoretic terms: analytically, it is the barycenter of the weight-measure associated to ; moduli-theoretically, it is the closure of the set of points where has semistable reduction, in the sense of geometric invariant theory.
"A new equivariant in nonarchimedean dynamics." Algebra Number Theory 11 (4) 841 - 884, 2017. https://doi.org/10.2140/ant.2017.11.841