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  "body": "\n## Definition\n\nLet $K$ be a field and $d \\ge 1$.\nAn affine hyperplane in $K^d$ is a set $\\set{x \\in K^d : a_1 x_1 + \\cdots + a_d x_d = b}$ with $(a_1, \\dots, a_d) \\ne 0$ and $b \\in K$; it is an [[affine subspace|0PYW26VNT6]] of dimension $d - 1$.\nA *hyperplane arrangement* in $K^d$ is a finite set $\\mathcal{A}$ of affine hyperplanes in $K^d$.\n\n## Remarks\n\nThe arrangement is *central* if all its hyperplanes have a common point.\nIts *intersection poset* $L(\\mathcal{A})$ is the set of all nonempty intersections of subsets of $\\mathcal{A}$, including $K^d$ itself as the empty intersection, [[partially ordered|0QV2JS9RJ3]] by reverse inclusion.\nFor $K = \\R$, the *regions* of $\\mathcal{A}$ are the connected components of $\\R^d \\setminus \\bigcup_{H \\in \\mathcal{A}} H$; each is an open convex set.\n\nExample: in $\\R^2$, the three lines $x = 0$, $y = 0$ and $x + y = 1$ form an arrangement.\nIt is not central, since the three lines have no common point.\nThey cut the plane into $7$ regions, one of them the bounded open triangle with vertices $(0,0)$, $(1,0)$ and $(0,1)$.\n\nNon-example: the unit circle together with the line $y = 0$ is not a hyperplane arrangement, since the circle is not a hyperplane.\nAn infinite family such as the lines $x = k$, $k \\in \\Z$, is not one either, since an arrangement is finite.\n",
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