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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $a = (a_1, \\dots, a_n) \\in \\C^n$ and let $r = (r_1, \\dots, r_n)$ with each $r_j \u003e 0$.\nThe *open polydisc* with centre $a$ and polyradius $r$ is\n$$P(a, r) = \\set{z \\in \\C^n : \\abs{z_j - a_j} \u003c r_j \\text{ for } j = 1, \\dots, n}.$$\nThe *closed polydisc* $\\overline{P}(a, r)$ is defined in the same way with $\\le$ in place of $\u003c$.\n\n## Remarks\n\nSo $P(a, r) = D(a_1, r_1) \\times \\cdots \\times D(a_n, r_n)$, a product of open discs in $\\C$.\nThe *unit polydisc* is $\\Delta^n = P(0, (1, \\dots, 1))$, the $n$-fold product of the [[unit disc|8G5ABXRK85]].\nExample: $(1/2, i/2) \\in \\Delta^2$, since both coordinates have modulus $1/2 \u003c 1$.\nNon-example: $(0, 1) \\notin \\Delta^2$, since the second coordinate has modulus $1$, which fails the strict inequality.\nThe open polydisc is the open ball of radius $1$ about $a$ for the norm $\\max_j \\abs{z_j - a_j}/r_j$.\nFor $n = 1$ a polydisc is an ordinary disc.\n",
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