{\mathbb{R}}^{n \times n}$, $b \in {\mathbb{R}}^m$ give the linear constraints (for notational simplicity we take all blocks to have the same size). Semidefinite programming is a broad generalization of linear programming (which we recover by setting $n=1$ in the above formulation), and, as for linear programming, there exist efficient algorithms for solving them. The reason semidefinite programming comes into play here, is that we can model polynomial inequality constraints as sum-of-squares constraints, which in turn can be written as semidefinite constraints; see, e.g., [@Bl].

Proof of Theorems \[thmNstar\], \[thmsimplezerosofdir\], and \[thmn1star\]
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To obtain the first part of Theorem \[thmNstar\] from Theorem \[thmsimplezerosofxi\] we need to minimize the functional ${\mathcal{Z}}$ over the space $\mathcal A_{LP}(d)$. We can see this as a bilevel optimization problem, where we optimize over scalars $R \geq 1$ in the outer problem, and over functions $f \in \mathcal A_{LP}(d)$ satisfying $r(f) = R$ in the inner problem. The outer problem is a simple one dimensional optimization problem for which we use Brent’s method [@Br]. The inner problem can be written as a semidefinite program as we discuss below. The numerical results suggest that the optimal $R$ goes to $1$ as $d \to \infty$ (which is itself 