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  "body": "\n## Statement\n\nLet $P \\subseteq \\R^n$ be a [[lattice polytope|F39N9NBFPZ]] of dimension $d$, and let $E_P$ be its [[Ehrhart polynomial|9DEG0WRDHR]].\nLet $P^\\circ$ be the relative interior of $P$, that is, its interior inside its affine hull.\nThen for every integer $t \\ge 1$,\n$$E_P(-t) = (-1)^d \\, \\#\\left(t P^\\circ \\cap \\Z^n\\right).$$\n\n## Remarks\n\nOne proof triangulates $P$ into lattice simplices and writes $P$ and $P^\\circ$ as disjoint unions of half-open simplices.\nFor a single simplex, the integer points of all dilates are counted with a rational generating function over the cone above the simplex.\nThe generating functions of a cone and of its interior are related by $z \\mapsto 1/z$, and this symmetry gives the sign $(-1)^d$ and the change from $P$ to $P^\\circ$.\n\nExample: for $P = [0,1]^2$, $E_P(t) = (t+1)^2$, so $E_P(-t) = (t-1)^2$.\nThe open square $(0,t)^2$ contains exactly the integer points $(a,b)$ with $1 \\le a, b \\le t-1$, and there are $(t-1)^2$ of them.\nFor the triangle with vertices $(0,0)$, $(1,0)$, $(0,1)$, $E_P(-t) = (t-1)(t-2)/2$, which counts the integer points with $a, b \\ge 1$ and $a + b \\le t - 1$.\n\nTaking $t = 1$ gives the number of interior lattice points of $P$ as $(-1)^d E_P(-1)$.\nThe statement extends to rational polytopes, where $E_P$ is replaced by the Ehrhart quasi-polynomial.\n\n## Explanations\n\n### L1 human\n\nThe Ehrhart polynomial counts grid points in a blown-up polytope, boundary included.\nPlugging negative numbers into the same polynomial counts the grid points strictly inside, up to a sign.\nSo one formula knows both the closed count and the open count.\n\n### L2 human\n\nReciprocity says that the Ehrhart polynomials of $P$ and of $P^\\circ$ are tied by $t \\mapsto -t$.\nIt is one instance of a family of combinatorial reciprocity laws, which also includes reciprocity for order polynomials and for chromatic polynomials.\nTogether with $E_P(0) = 1$ and the leading coefficient being the volume, it constrains the coefficients of $E_P$.\nFor example, a lattice polygon $P$ with area $A$ and $b$ lattice points on its boundary has $E_P(t) = A t^2 + \\frac{b}{2} t + 1$.\nReciprocity at $t = 1$ then gives $A - \\frac{b}{2} + 1$ interior lattice points, which is Pick's theorem.\n",
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  "msc": [
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  "name": "Ehrhart–Macdonald reciprocity",
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