The Bohr radius $R$ for $\mathcal{H}(\mathbb{D})$ is defined as the radius $$R = \sup\limits_{0<r<1} \Bigl\{ r\ \Big|\sum\limits_{k=0}^{\infty}|a_k|r^k \leq |f|_\mathbb{D}, \text{ for all }f(z)=\sum\limits_{k=0}^{\infty} a_{k}z^{k}\in\mathcal{H}(\mathbb{D}) \Bigr\},$$ where $|f|_{\mathbb{D}}=\sup\limits_{z\in\mathbb{D}} |f(z)|$, $\mathcal{H}(\mathbb{D})$ is the space of functions analytic in $\mathbb{D}$, and $\sum\limits_{k=0}^{\infty}|a_k|r^k$ is known as the majorization of $f$. Harald Bohr and later M. Riesz, I. Shur, and F. Wiener showed that $R=\frac{1}{3}$, and it is known the only extremal functions satisfying $\sum\limits_{k=0}^\infty |a_k|(\frac{1}{3})^k=|f|_{\mathbb{D}}$ are constant. There are several extensions of this result. For instance one can exactly compute the radius in terms of entries in an associated Toeplitz matrix when $\mathcal{H}(\mathbb{D})$ is replaced with $\mathcal{P}_{n}$, polynomials of degree of at most $n$. Another interesting case is for functions on the bi-disk $f\in\mathcal{H}(\mathbb{D}^2)$, where the radius is unknown.
One interesting question is by which means could one effectively numerically estimate the Bohr radius for the space of polynomials in $n$ variables of degree at most $m$. More precisely, the following problem is interesting.
Problem: Devise means to numerically, or better yet exactly, compute a sequence of polynomials $p_{k}(z_1,\dots,z_n)$ each of degree at most $m$, so that the majorization of $p_{k}(z_1,\dots,z_n)$, denoted $p^{*}_{k}$, attains the maximum value of $|p_{k}|$ on the poly-disk $\mathbb{D}^n$ at $p^{*}_{k}(r,r,...,r)$ for value $r=r_k$, and $r_k\rightarrow \mathcal{R}^{m,n}$, as $k\rightarrow \infty$, where $\mathcal{R}^{m,n}$ is the corresponding Bohr radius, given as the supremum over all $r$ satisfying $$ \Bigl|\sum\limits_{|\alpha|\leq m} c_{\alpha}z^{\alpha}\Bigr|<1\ \text{ for }\ (z_1,\dots,z_{n})\ \text{ s.t. }\ \|z\|_{\infty}=\max\limits_{1\leq j\leq n} |z_{j}|<1\ \text{ implies} \\ \sum\limits_{|\alpha|\leq m} |c_{\alpha}z^{\alpha}|<1\ \text{for } \|z\|_{\infty}<r$$
Edit: For anyone interested, some lower bounds for the Bohr radius in multiple variables are easily derived following a Cauchy-Schwarz argument as in H. P. Boas and D. Khavinson's paper (which gave first lower and upper bounds for the Bohr radius on the poly-disk). For two variable polynomials of degree $2,3,$ and $4$ respectively the bounds one get are $\geq.26654,\geq.25261,$ and $\geq.24936$. For three variables one gets for degree $2,3,$ and $4$ the lower bounds $\geq.22014,\geq.20967,$ and $\geq.20741$.
