  We study convergence rates for Gibbs measures, with density proportional to
$e^{-f(x)/t}$, as $t \rightarrow 0$ where $f : \mathbb{R}^d \rightarrow
\mathbb{R}$ admits a unique global minimum at $x^\star$. We focus on the case
where the Hessian is not definite at $x^\star$. We assume instead that the
minimum is strictly polynomial and give a higher order nested expansion of $f$
at $x^\star$, which depends on every coordinate. We give an algorithm yielding
such a decomposition if the polynomial order of $x^\star$ is no more than $8$,
in connection with Hilbert's $17^{\text{th}}$ problem. However, we prove that
the case where the order is $10$ or higher is fundamentally different and that
further assumptions are needed. We then give the rate of convergence of Gibbs
measures using this expansion. Finally we adapt our results to the multiple
well case.
