On solutions of linear equations with polynomial coefficients

TL;DR

This paper investigates the existence of solutions with different regularity properties for linear functional equations with polynomial coefficients, revealing that certain solutions do not imply others with higher regularity.

Contribution

It demonstrates that solutions with higher regularity (arc-analytic, Nash regulous) do not necessarily exist even when continuous semialgebraic solutions do, highlighting limitations in solution regularity.

Findings

Arc-analytic solutions may not exist despite continuous semialgebraic solutions.

Nash regulous solutions may not exist despite arc-analytic solutions.

Regularity properties of solutions are not necessarily preserved across different solution classes.

Abstract

We show that a linear functional equation with polynomial coefficients need not admit an arc-analytic solution even if it admits a continuous semialgebraic one. We also show that such an equation need not admit a Nash regulous solution even if it admits an arc-analytic one.