On the analog of the Kolmogorov-Arnold superposition representation for continuous functions of several $p$-adic variables

TL;DR

This paper addresses extending the Kolmogorov-Arnold superposition principle to functions of several $p$-adic variables. It proves two main results: Theorem 1 provides a simple univariate superposition representation for continuous real-valued functions on $\mathbb{Z}_p^n$, and Theorem 2 establishes a similar univariate superposition for continuous $\mathbb{Q}_p$-valued functions. The proofs for both cases rely on explicit $p$-adic constructions, including Cantor-type encodings, homeomorphisms, and, in the real-valued case, the Tietze extension theorem, with the $p$-adic case using the structure of homomorphisms $\Phi(x,y)=\omega(x)+p\omega(y)$. These results demonstrate that multivariate dependence on $p$-adic variables reduces to univariate compositions, grounded in canonical $p$-adic decompositions. The findings have potential implications for $p$-adic mathematical physics and network architectures inspired by Kolmogorov-Arnol d-type representations by simplifying function approximation in non-Archimedean settings.

Abstract

It is shown that any continuous function depending on several $p$-adic variables, each of which is defined on $\mathbb{Z}_{p}$, can be represented as a superposition of continuous functions of one $p$-adic variable. This statement is true for both functions with values in $\mathbb{R}$ and functions with values in $\mathbb{Q}_{p}$.