  Let $\Omega_0$ be a polygon in $\RR^2$, or more generally a compact surface
with piecewise smooth boundary and corners. Suppose that $\Omega_\e$ is a
family of surfaces with $\calC^\infty$ boundary which converges to $\Omega_0$
smoothly away from the corners, and in a precise way at the vertices to be
described in the paper. Fedosov \cite{Fe}, Kac \cite{K} and McKean-Singer
\cite{MS} recognized that certain heat trace coefficients, in particular the
coefficient of $t^0$, are not continuous as $\e \searrow 0$. We describe this
anomaly using renormalized heat invariants of an auxiliary smooth domain $Z$
which models the corner formation. The result applies both for Dirichlet and
Neumann conditions. We also include a discussion of what one might expect in
higher dimensions.
