Projected Brion's theorem
Abstract
We present an analogue of Brion's theorem for the projected integer-point transform. Brion's theorem writes the integer-point transform of a rational polyhedron P as the sum of the rational functions associated to its vertex cones. The pushforward π* along a lattice projection π turns the transform of P into the generating function of the numbers of lattice points in the fibres of π. We group the vertices of P into contracted classes, generated by the edges whose direction π kills, and show that the sum of the rational functions associated to the vertex cones in a contracted class can be pushed forward along π, yielding a rational function decomposition of the projected integer-point transform. For a lattice polytope, the classes correspond to the connected components of the fixed locus of the associated toric variety under the action of a subtorus, and our formula resembles the localization formula for this subtorus. We use this decomposition to provide an explicit formula for the projected integer-point transform of a linear family of polytopes. We illustrate this result with families of Gelfand--Zetlin and string polytopes.
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BibTeX
@misc{dupraz2026projected,
title = {{Projected Brion's theorem}},
author = {Matthew Dupraz and Leonid Monin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38674},
eprint = {2610.38674},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38674},
}
RIS
TY - UNPB
TI - Projected Brion's theorem
AU - Matthew Dupraz
AU - Leonid Monin
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38674
UR - https://arxiv.org/abs/2610.38674
AB - We present an analogue of Brion's theorem for the projected integer-point transform. Brion's theorem writes the integer-point transform of a rational polyhedron P as the sum of the rational functions associated to its vertex cones. The pushforward π_* along a lattice projection π turns the transform of P into the generating function of the numbers of lattice points in the fibres of π. We group the vertices of P into contracted classes, generated by the edges whose direction π kills, and show that the sum of the rational functions associated to the vertex cones in a contracted class can be pushed forward along π, yielding a rational function decomposition of the projected integer-point transform. For a lattice polytope, the classes correspond to the connected components of the fixed locus of the associated toric variety under the action of a subtorus, and our formula resembles the localization formula for this subtorus. We use this decomposition to provide an explicit formula for the projected integer-point transform of a linear family of polytopes. We illustrate this result with families of Gelfand--Zetlin and string polytopes.
ER -