Lemma: Uniqueness of polynomial kernels

The symmetric kernels in (eq:polynomial) are uniquely determined by the functional on all smooth configurations. Consequently ${\mathcal{P}}_{\mathrm{dist}}$ may be identified with its coefficient subspace in ${\mathcal{T}}_{\mathrm{coeff}}$.

Proof. Scaling a configuration separates homogeneous degrees. Polarization of a homogeneous polynomial recovers its symmetric multilinear form. If the polynomial is zero, that multilinear form vanishes on every tuple of smooth configurations.

In particular it vanishes on compactly supported elementary tensor tests in each bundle component. Finite sums of such tensor tests are dense in the smooth test space on each compact product region. Continuity of the distribution therefore makes the corresponding kernel zero.