the entries of $X_3$ and $X_4$. Finally, the objective ${\mathcal{Z}}(f)$ is a linear combination in the entries of $X_1$ and $X_2$, which can be implemented by using the identity $$\int x^m e^{-\pi x^2} {\,\text{\rm d}}x = -\frac{1}{2\pi^{m/2+1/2}} \Gamma\Big(\frac{m+1}{2}, \pi x^2\Big),$$ where $\Gamma$ is the upper incomplete gamma function. Hence, the problem of minimizing ${\mathcal{Z}}(f)$ over functions $f \in \mathcal A_{LP}(d)$ that satisfy $r(f) = R$ is a semidefinite program.

To obtain the second part of Theorem \[thmNstar\] from Theorem \[thmsimplezerosofxi\] and to obtain Theorem \[thmsimplezerosofdir\] from \[thmnphi\] we use the same approach with a different functional. To obtain Theorem \[thmn1star\] from Theorem \[thmsimplezerosofxiprime\] we also do the same as above, but now truncate the series in the functional ${\mathcal{Z}}_{1}$ at $k=15$ and add the easy to compute upper bound $10^{-10}$ on the remainder of the terms.

### Implementation and numerical issues {#sec:rig}

In implementing the above as a semidefinite program we have to make two choices for the polynomial basis that we use: the basis defining the vector $v(x)$, and the basis to enforce the identity $I(X_1,\ldots,X_4) = 0$. This choice of bases is important for the numerical conditioning of the resulting semidefinite program. Following [@dLOV] we choose the Laguerre 