  Consider an algorithm computing in a differential field with several
commuting derivations such that the only operations it performs with the
elements of the field are arithmetic operations, differentiation, and zero
testing. We show that, if the algorithm is guaranteed to terminate on every
input, then there is a computable upper bound for the size of the output of the
algorithm in terms of the size of the input. We also generalize this to
algorithms working with models of good enough theories (including for example,
difference fields).
  We then apply this to differential algebraic geometry to show that there
exists a computable uniform upper bound for the number of components of any
variety defined by a system of polynomial PDEs. We then use this bound to show
the existence of a computable uniform upper bound for the elimination problem
in systems of polynomial PDEs with delays.
