Continuous rational functions on real and $p$-adic varieties II

TL;DR

This paper investigates the properties of continuous rational functions on real and p-adic algebraic varieties, focusing on restriction, extension, and solvability of linear equations, advancing understanding in algebraic geometry.

Contribution

It provides partial solutions to fundamental questions about the behavior and applicability of continuous rational functions on algebraic varieties over real and p-adic fields.

Findings

Restrictions of functions to subvarieties are characterized.

Extensions of functions from subvarieties are analyzed.

Linear equations solvability using continuous rational functions is explored.

Abstract

This note replaces two earlier preprints (1101.3737 by Koll\'ar) and (1211.6681 by Nowak). It studies, and partially solves, 3 elementary questions about continuous rational functions on real (and p-adic) algebraic varieties: Can one restrict such a function to a subvariety? Can one extend such a function from a subvariety? Which linear equations can be solved using such functions? v.2: Section 3 simplified and typos corrected.