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  "body": "\n## Definition\n\nLet $P \\subseteq \\R^n$ be a [[lattice polytope|F39N9NBFPZ]] of dimension $d$.\nFor each integer $t \\ge 1$, let\n$$L_P(t) = \\#\\left(tP \\cap \\Z^n\\right), \\qquad tP = \\set{tx : x \\in P},$$\nbe the number of integer points in the dilate $tP$.\nBy Ehrhart's theorem, there is a unique [[polynomial|1G0B73WPX8]] $E_P(t)$ with rational coefficients such that $E_P(t) = L_P(t)$ for all integers $t \\ge 1$.\nThis polynomial $E_P$ is the *Ehrhart polynomial* of $P$.\n\n## Remarks\n\nThe polynomial $E_P$ has degree $d$, and $E_P(0) = 1$.\nIf $d = n$, the leading coefficient of $E_P$ is the volume of $P$.\nIn general the leading coefficient is the volume of $P$ measured inside its affine hull, normalised by the lattice of integer points of that affine hull.\n\nExample: for the unit square $P = [0,1]^2$, the dilate $tP = [0,t]^2$ contains $(t+1)^2$ integer points, so $E_P(t) = t^2 + 2t + 1$.\nFor the triangle with vertices $(0,0)$, $(1,0)$, $(0,1)$, the dilate contains the points $(a,b) \\in \\Z^2$ with $a, b \\ge 0$ and $a + b \\le t$, so $E_P(t) = (t+1)(t+2)/2$.\nIn both cases the leading coefficient is the area, $1$ and $1/2$.\n\nNon-example: the segment $P = [0, 1/2]$ is a rational polytope but not a lattice polytope.\nHere $L_P(t) = \\floor{t/2} + 1$, which is not a polynomial in $t$; it is only a quasi-polynomial of period $2$.\nSo the hypothesis that the vertices are integer points is needed.\n\nThe generating function $\\sum_{t \\ge 0} E_P(t) z^t$ is a rational function of the form $h^*(z)/(1-z)^{d+1}$, where $h^*$ is a polynomial with nonnegative integer coefficients.\n",
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