Carneiro and Vaaler have proved, as an application of their work on Beurling-Selberg extremal majorants, that for any non-zero complex polynomial $f(z) \in \mathbb{C}[z]$, the infimum value of the integral $c_0$ of a degree-$N$ symmetric Laurent polynomial majorant $P_{N,f}(z) = \sum_{k = -N}^N c_k z^k$ of the logarithmic function $$ \log{|f|}\leq P_{N,f} \quad \textrm{ on the unit circle } \mathbb{T} $$ satisfies $$ \inf_{P_{N,f}} \, \int_{\mathbb{T}} P_{N,f}(z) \, d\mu_{\mathrm{Haar}}(z) \leq \int_{\mathbb{T}} \log{|f|} \, d\mu_{\mathrm{Haar}} + \frac{\log{2}}{N+1} \cdot \deg{f}, $$ which in the fundamental case $f(z) = 1 - z$ is an equality and attained for a unique degree-$N$ symmetric Laurent polynomial $P_N$. (The first two of those extremal polynomials are $P_0(z) = \log{2}$ and $P_1(z) = \frac{1}{2}( \log{2} - (z + z^{-1})/2 )$.) The case of an arbitrary $f$ gets straightforwardly reduced to the $f(z) = 1 - z$ case, but this way of phrasing the result suggests an immediate generalization:

I would like to raise the following for multivariate polynomials.

Problem. Let $f \in \mathbb{C}[z_1^{\pm 1}, \ldots, z_d^{\pm d}] \setminus \{0\}$, a non-zero Laurent polynomial in $d$ commuting variables. For a given $d$-tuple $(N_1,\ldots,N_d)$ of degrees, find the infimum value (or its decent upper bound as above) of $\int_{\mathbb{T}^d} P(\mathbf{z}) \, d\mu_{\mathrm{Haar}}(\mathbf{z}) = c_{0,\ldots,0}$ over all symmetric Laurent polynomial majorants $$ \log{|f(\mathbf{z})|} \leq P(\mathbf{z}) = \sum_{k_1 = -N_1}^{N_1} \cdots \sum_{k_d = -N_d}^{N_d} c_{k_1,\ldots,k_d} z_1^{k_1} \cdots z_d^{k_d} \, \textrm{ on the unit torus } \mathbb{T}^d $$ with multi-degree $\preceq (N_1, \ldots, N_d)$. Does it converge to the logarithmic Mahler measure of $f$ (the integral of the dominated function), as $\min(N_1,\ldots,N_d) \to \infty$? Does it do so at the rate $O_f(1/N_1 + \cdots + 1/N_d)$, like it does in the univariate case?

Note. "Symmetric" means that the Laurent polynomial is stable under the involution mapping $z_i \mapsto 1/z_i$ for $i = 1,\ldots,d$. This ensures that $P(\mathbf{z})$ takes real values on the unit torus $\mathbb{T}^d$, and so the inequalities make sense.


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