Scattering diagrams: polynomiality and the dense region

Abstract

We use deformations and mutations of scattering diagrams to show that the coefficients of a scattering diagram with initial functions $f1 = (1+tx)^{\mu}$ and $f2 = (1+ty)^{\nu}$ are polynomial in ${\mu}$, ${\nu}$ and non-trivial in a certain dense region. We discuss consequences for Gromov-Witten invariants and quiver representations.