The study of nonlinear PDEs relies on a priori estimates, 
here called entropy-entropy production inequalities, which 
can be derived from appropriate integration by parts. 
However, these integration by parts can be sometimes quite
involved. In this talk we present a systematic method
for deriving such a priori estimates for a large class of 
nonlinear even-order PDEs in one or several space dimensions 
with periodic boundary conditions. This class also contains 
the thin-film equation.

The main new idea is to identify the rules of integration by 
parts with suitable polynomial manipulations. In this way,
the non-negativity of the entropy production term (leading
to an a priori estimate) is equivalent to a decision problem
from real algebraic geometry which can be solved by computer-
aided quantifier elimination. We show that our method is also 
able to treat compound equations with different orders, to 
prove non-existence of entropies, and to derive new logarithmic 
Sobolev-type inequalities.