  We initiate a systematic study of nonnegative polynomials $P$ such that $P^k$
is not a sum of squares for any odd $k\geq 1$, calling such $P$
\emph{stubborn}. We develop a new invariant of a real isolated zero of a
nonnegative polynomial in the plane, that we call \emph{the SOS-invariant}, and
relate it to the well-known delta invariant of a plane curve singularity. Using
the SOS-invariant we show that any polynomial that spans an extreme ray of the
convex cone of nonnegative ternary forms of degree 6 is stubborn. We also show
how to use the SOS-invariant to prove stubbornness of ternary forms in higher
degree. Furthermore, we prove that in a given degree and number of variables,
nonnegative polynomials that are not stubborn form a convex cone, whose
interior consists of all strictly positive polynomials.
