# Extremal polynomial majorants of $\log{|f|}$: a multivariate extension of a theorem of Carneiro and Vaaler

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.