Ehrhart reciprocity
= Ehrhart reciprocity
{c}
{title2=$L_{P^\circ}(n)=(-1)^dL_P(-n)$}
For a $d$-dimensional <lattice polytope> $P$ and <positive integer> $n$, <Ehrhart reciprocity> states
$$
\#(nP^\circ\cap\mathbb Z^d)=(-1)^dL_P(-n),
$$
with <relative interior> when needed. Negative input means <polynomial evaluation>, not a negative dilation count. For the <unit square>, $(1-n)^2=(n-1)^2$ counts the interior grid points of its $n$th dilation.<Computing the Continuous Discretely by Matthias Beck and Sinai Robins>[§4.3, Theorem 4.1]{ref}