Exponential valuations on lattice polygons valued at formal power series

TL;DR

This work classifies $\mathcal{G}(\mathbb Z^2)$-equivariant valuations on lattice polygons with values in $\mathbb Q[[x,y]]$ via a dilative decomposition indexed by integers $\delta\ge -2$, and provides a constructive parametrization by constants, a univariate power series, and a $D_4$-invariant bivariate series. Central to the method is a bijection between simple valuations and a $D_4$-invariant subspace $\mathcal V$ of power series, given explicitly by $Z(T)=\frac{e^{x}-e^{y}}{(x-y)y}\,\varrho(y-x,x)-\frac{e^{x}-1}{xy}\,\varrho(x,y-x)$. The results reveal that the even-$\delta$ sector aligns with modular-forms dimensions, and yield explicit dilative parameters, including Laplace-transform-type examples, while providing a practical tool to generate and recognize all equivariant valuations. The framework extends prior work by Freyer–Ludwig–Rubey and Li et al., offering a constructive route to both synthesize and classify these valuations in the lattice-polygon setting.

Abstract

We classify valuations on lattice polygons with values in the ring of formal power series that commute with the action of the affine unimodular group. A typical example of such valuations is induced by the Laplace transform, but as it turns out there are many more. The classification is done in terms of formal power series that satisfy certain functional equations. We align our classification with the decomposition into so-called dilative components.

Theorems & Definitions (37)

• Theorem 1: Li, Ma LiM17
• Theorem 2
• Theorem 3: Freyer, Ludwig, Rubey FMR
• Theorem 4
• Remark 5
• Proposition 6
• Lemma 7
• Proposition 8
• Corollary 9
• Lemma 10