This work classifies valuations that commute under the affine unimodular group, suggesting a deeper understanding of functional equations.
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.
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Böröczky et al. (2025) studied this question.
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