Exponential valuations on lattice polygons

TL;DR

This work classifies translatively exponential and ${\rm GL}(2,\mathbb Z)$ covariant valuations on lattice polygons valued in measurable real functions. By reducing to the data on the origin, a lattice edge, and the basic triangle, and leveraging ergodicity of the ${\rm SL}(2,\mathbb Z)$ action, the authors derive a parametric framework in which valuations are determined by $f_0=f_0$, $f_1=f_1$, and $f_2=f_2$ living in ${\mathcal M}(\mathbb R^2)^{GL(2,\mathbb Z)}$, ${\mathcal M}(\widetilde{\Omega}_{1})$, and ${\mathcal M}(\widetilde{\Omega}_{2})$ respectively, subject to explicit functional equations. They provide constructive building blocks, including the positive Laplace transform and an additional simple valuation ${Z_1}$, and establish a geometric method (triangulations and Fibonacci-based partitions) to extend local data to all lattice polygons. The main result yields a complete decomposition ${Z=Z_1+Z_2}$ with ${Z_2}$ simple and ${Z_1}$ capturing low-dimensional contributions, and proves the existence of non-Laplace analytic valuations, enriching the landscape of lattice polytope valuations with exponential weights. This blends ergodic theory, combinatorics, and analysis to advance intrinsic valuation theory on lattice polytopes.

Abstract

We classify translatively exponential and GL(2,Z) covariant valuations on lattice polygons valued at measurable real functions. A typical example of such valuations is induced by the Laplace transform, but as it turns out there are many more. The argument uses the ergodicity of the linear action of SL(2,Z) on R2, and some elementary properties of the Fibonacci numbers.

Figures (2)

• Figure 1: The domain $\widetilde{\Omega}_2$.
• Figure 2: The domain $\widetilde{\Omega}_1$.

Theorems & Definitions (24)

• Theorem 1: Li, Ma LiM17
• Conjecture 2
• Theorem 3
• Proposition 4
• Theorem 5
• Lemma 6
• Lemma 7
• Proposition 8
• Lemma 9
• Example 10: Positive Laplace transform