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I would like to calculate the set of $z\in \mathbb{C}^d$ such that there exists a constant $C >0$ such that for every homogeneous polynomial $p$ in $d$ variables $$|p(z)|\leq C\sup_{x\in [0,1]^d} |p(x)|.$$ Or at least the interior of this set. I can show that it contains a polydisk of radius $1/1000$ or so, but I assume better bounds are attainable or known. That is, the described set contains the $z\in \mathbb{C}^d$ such that each $|z_i|< 1/1000.$

So specifically, is there an exact description of this set?

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