The dynamical uniform boundedness conjecture asserts a uniform bound for the number of preperiodic points of fixed-degree morphisms
of projective space over number
fields (Morton and Silverman 1994; Silverman 2007, p. 95). Fix integers 
,
,
and 
. There is a constant 
such that, for every number field 
with extension field
degree ![[K:Q]<=D](/images/equations/DynamicalUniformBoundednessConjecture/Inline6.svg)
and every morphism 
of degree 
defined over 
,
 |
where 
is the set of preperiodic
points in 
.
Here 
is the common total degree
of the homogeneous polynomials giving 
in homogeneous coordinates, with no common
zero in projective space
over an algebraic closure of 
. The set of preperiodic
points for each individual 
over 
is known to be finite. The conjecture
asks for a bound that is independent of 
and of the choice of 
within the specified extension
field degree bound (Silverman 2007, p. 95).
