Linear equations on real algebraic surfaces

TL;DR

This paper proves that for nonsingular real algebraic surfaces, any continuous solution to a linear equation with continuous rational coefficients can be replaced by a continuous rational solution, highlighting a special property in two dimensions.

Contribution

It establishes a dimension-specific result showing the equivalence of continuous and continuous rational solutions for linear equations on real algebraic surfaces.

Findings

Continuous solutions imply continuous rational solutions on surfaces.

The result does not extend to higher-dimensional algebraic varieties.

Provides insight into the structure of solutions on real algebraic surfaces.

Abstract

We prove that if a linear equation, whose coefficients are continuous rational functions on a nonsingular real algebraic surface, has a continuous solution, then it also has a continuous rational solution. This is known to fail in higher dimensions.