Year: 2026
Type: preprint
Abstract: In this paper we study definable families of functions from an ordered abelian group into various naturally arising definable quotients. We show that for an ordered abelian group $G$ and definable family of convex subgroups $\{D\}_{D\in\mathcal{D}}$, any definable family of functions $\{f_D\} _{D\in\mathcal{D}}$ with $f_D:G^d\rightarrow\frac{G}{D}$ is uniformly piecewise linear; for a prime $p$, integers $s,r\geq 1$, and groups $D^{[p^s]}$ defined later, if $f_D:G^d\rightarrow\frac{G}{D+p^rG}$ or $f_D:G^d\rightarrow\frac{G}{D^{[p^s]}+p^rG}$ we instead obtain that the definable family of functions is uniformly piecewise a boolean combination of linear functions to quotients by subgroups which are uniformly definable from $D$.
Source: arXiv (Cornell University)
Author: Wells, Harper
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7148350748
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