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In this paper, we make partial progress on a function field version of the dynamical uniform boundedness conjecture for certain one-dimensional families
of polynomial maps, such as the family
. We do this by making use of the dynatomic modular curves
) which parametrize maps
together with a point (respectively orbit) of period
. The key point in our strategy is to study the set of primes
for which the reduction of
fails to be smooth or irreducible. Morton gave an algorithm to construct, for each
, a discriminant
whose list of prime factors contains all the primes of bad reduction for
. In this paper, we refine and strengthen Morton’s results. Specifically, we exhibit two criteria on a prime
: one guarantees that
is in fact a prime of bad reduction for
, yet this same criterion implies that
is geometrically irreducible. The other guarantees that the reduction of
is actually smooth. As an application of the second criterion, we extend results of Morton, Flynn, Poonen, Schaefer, and Stoll by giving new examples of good reduction of
for several primes dividing
. The proofs involve a blend of arithmetic and complex dynamics, reduction theory for curves, ramification theory, and the combinatorics of the Mandelbrot set.