[Paper Review] Reconstruction of rational polytopes from the real-parameter Ehrhart function of its translates

Tiago Royer|arXiv (Cornell University)|Dec 5, 2017

Advanced Combinatorial Mathematics1 references3 citations

TL;DR

This paper establishes that a rational polytope is uniquely determined by the real-parameter Ehrhart functions of all its integer translates. By analyzing the growth and discontinuity behavior of these functions, the authors prove that if two rational polytopes have identical Ehrhart functions across all integer translations, they must be identical, not just unimodularly equivalent. This provides a complete reconstruction method for rational polytopes from their extended Ehrhart functions.

ABSTRACT

When extending the Ehrhart lattice point enumerator $L_P(t)$ to allow real dilation parameters $t$, we lose the invariance under integer translations that exists when $t$ is restricted to be an integer. This paper studies this phenomenon; in particular, it is shown that, for full-dimensional $P$, not only there are infinitely many different functions $L_{P + w}(t)$ (for integer $w$), but that for rational $P$ the collection of these functions identifies $P$ uniquely.

Motivation & Objective

• To investigate whether the real-parameter Ehrhart function of a rational polytope's integer translates uniquely determines the polytope.
• To extend classical Ehrhart theory, which is invariant under integer translations for integer dilations, to real dilation parameters where such invariance breaks.
• To determine whether the full collection of Ehrhart functions $ L_{P+w}(s) $ for all integer vectors $ w $ uniquely identifies a rational polytope $ P $.
• To explore the possibility of finite witness sets for reconstruction and extend results to semi-rational and real polytopes.

Proposed method

• Analyzing the behavior of $ L_{P+w}(s) $ as a function of real $ s > 0 $, particularly its discontinuities and growth rates.
• Using the fact that $ L_{P}(s) $ is nondecreasing if and only if $ 0 \in P $, to distinguish polytopes based on translation effects.
• Applying volume growth analysis of pseudopyramids $ \operatorname{ppyr}(P+kw) $ to detect differences in polytope structure under translation.
• Projecting polytopes onto lower-dimensional subspaces via unimodular transformations to reduce dimension and apply induction.
• Using dense sets of rational $ s $-values (e.g., $ s = m/(b + w_d) $) to preserve function equality across projections.
• Proving uniqueness by contradiction: assuming $ P \neq Q $, showing that their Ehrhart functions must differ for some $ w $.

Experimental results

Research questions

• RQ1 Can a rational polytope be uniquely reconstructed from the real-parameter Ehrhart functions of all its integer translates?
• RQ2 Does the loss of invariance under integer translation in the real-parameter Ehrhart function allow for distinguishing different polytopes?
• RQ3 Is there a finite set of integer translation vectors $ w $ such that the Ehrhart functions $ L_{P+w}(s) $ uniquely determine $ P $?
• RQ4 Can the reconstruction result be extended to semi-rational or general real polytopes?

Key findings

• For any rational polytope $ P $, there exists an integer vector $ w $ such that the functions $ L_{P+kw}(s) $ are all distinct for $ k \geq 0 $, showing that translation breaks invariance in the real-parameter Ehrhart function.
• If two rational polytopes $ P $ and $ Q $ satisfy $ L_{P+w}(s) = L_{Q+w}(s) $ for all integer vectors $ w $ and all real $ s > 0 $, then $ P = Q $, proving unique reconstruction.
• The result extends to semi-rational polytopes (defined by integer normal vectors and real offsets) of full dimension or codimension one, under the same function equality condition.
• For full-dimensional real polytopes, the same conclusion holds under the same function equality condition, though the proof requires codimension assumptions.
• The proof relies on detecting structural differences via discontinuity locations and relative volume growth of pseudopyramids under translation.
• The authors conjecture that the full integer lattice $ \mathbb{Z}^d $ can be replaced by a finite witness set for rational polytopes, though this remains open.