  We present a method for solving polynomial equations over idempotent
omega-continuous semirings. The idea is to iterate over the semiring of
functions rather than the semiring of interest, and only evaluate when needed.
The key operation is substitution. In the initial step, we compute a linear
completion of the system of equations that exhaustively inserts the equations
into one another. With functions as approximants, the following steps insert
the current approximant into itself. Since the iteration improves its precision
by substitution rather than computation we named it Munchausen, after the
fictional baron that pulled himself out of a swamp by his own hair. The first
result shows that an evaluation of the n-th Munchausen approximant coincides
with the 2^n-th Newton approximant. Second, we show how to compute linear
completions with standard techniques from automata theory. In particular, we
are not bound to (but can use) the notion of differentials prominent in Newton
iteration.
