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We introduce a natural weighted enumeration of lattice points in a polytope,and give a Brion-type formula for the corresponding generating function. Theweighting has combinatorial significance, and its generating function may beviewed as a generalization of the Rogers-Szeg\H{o} polynomials. It also arisesfrom the geometry of the toric arc scheme associated to the normal fan of thepolytope. We show that the asymptotic behavior of the coefficients at $q = 1$is Gaussian.

Toric arc schemes and $q$-enumeration of lattice points