high enough in relation to the degree $d$. Here we explain the procedure we use to obtain bounds that are guaranteed to be correct. This is an adaptation of the method from [@Lo] and [@dLOV].

We first solve the above optimization problem numerically to find $R$ and $f$ for which we have a good objective value $v = {\mathcal{L}}(f)$. Then we solve the semidefinite program again for the same value of $R$, but now we solve it as a feasibility problem with the additional constraint ${\mathcal{L}}(f) \leq v + 10^{-6}$. The interior point solver will try to give the analytical center of the semidefinite program, so that typically the matrices are all positive definite; that is, the eigenvalues are all strictly positive. Then we use interval arithmetic to check rigorously that $X_2$, $X_3$, and $X_4$ are positive definite, and we compute a rigorous lower bound $b$ on the smallest eigenvalues of $X_3$ and $X_4$.

Using interval arithmetic we compute an upper bound $B$ on the largest coefficient of $I(0, X_2, X_3,X_4)$ in the basis given by the $2d+1$ entries on the diagonal and upper diagonal of the matrix $(R^2 - x^2) v(x^2) v(x^2)^{\sf T}$. If $b \geq (1+2d)B$, then it 