Semialgebraic and Continuous Solution of Linear Equations with Semialgebraic Coefficients

TL;DR

This paper extends previous work on linear equations with polynomial coefficients to semialgebraic coefficients, providing a necessary and sufficient condition for the existence of continuous semialgebraic solutions using Glaeser refinement techniques.

Contribution

It introduces a new approach based on algebraic geometry and Glaeser refinement to characterize solutions for systems with semialgebraic coefficients, generalizing prior polynomial-focused results.

Findings

A continuous semialgebraic solution exists if and only if the associated Glaeser-stable bundle has no empty fiber.

The result generalizes previous polynomial coefficient cases to semialgebraic coefficients.

Provides a necessary and sufficient condition for the existence of solutions in the semialgebraic setting.

Abstract

Starting from the results of Charles Fefferman and Janos Koll\'ar in \texit{Continuous Solutions of Linear Equations} [1], we adopt a new approach based on Fefferman's techniques of Glaeser refinement to show a more general result than the one proved by Koll\'ar by using techniques from algebraic geometry. Considering a system of linear equations with semialgebraic (not only polynomial as in [1]) coefficients on $\mathbb{R}^{n}$, we get a necessary and sufficient condition for the existence of a continuous and semialgebraic solution on $\mathbb{R}^{n}$. This is different from what Fefferman and Luli obtained in \textit{Semialgebraic Sections Over the Plane} since they stated their result for solutions of regularity $C^m$ on the plane $\mathbb{R}^2$. More in depth, we prove that a continuous and semialgebraic solution on $\mathbb{R}^{n}$ exists if and only if there is a continuous solution i.e., if the Glaeser-stable bundle associated to the system has no empty fiber.