  We study the minimal resultant divisor of self-maps of the projective line
over a number field or a function field and its relation to the conductor. The
guiding focus is the exploration of a dynamical analog to Theorem 1.1, which
bounds the degree of the minimal discriminant of an elliptic surface in terms
of the conductor. We study minimality and semi-stability, considering what
conditions imply minimality (Theorem 4.4) and whether semi-stable models and
presentations are minimal, proving results in the degree two case (Theorems
4.6, 4.7). We prove the singular reduction of a semi-stable presentation
coincides with the bad reduction (Theorem 3.1). Given an elliptic curve over a
function field with semi-stable bad reduction, we show the associated Lattes
map has unstable bad reduction (Theorem 3.6). Degree 2 maps in normal form with
semi-stable bad reduction are used to construct a counterexample (Theorem 2.1)
to a natural dynamical analog to Theorem 1.1. Finally, we consider the notion
of "critical bad reduction," and show that a dynamical analog to Theorem 1.1
may still be possible using the locus of critical bad reduction to define the
conductor (Theorem 4.10, Theorem 4.13).
