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

TL;DR

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

Contribution

It provides partial solutions to three basic questions about the behavior and applicability of continuous rational functions on these varieties.

Findings

Restrictions of functions to subvarieties are characterized.

Conditions for extending functions from subvarieties are identified.

Linear equations solvable via continuous rational functions are described.

Abstract

This note 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?