Definable Continuous Solutions of Linear Equations

Abstract

In this paper, we study a generalization of a question, raised by C. Fefferman and J. Kollár, on the existence of solutions of linear functional equations. Suppose that R is a definably complete expansion of a real closed field (R;+,⋅). Let f,g1,…,gk:Rn→R be continuous functions that are definable in R. We prove that if there exist continuous functions y1,…,yk:Rn→R such that f=g1y1+⋯+gkyk, then there exist continuous functions y1,…,yk such that y1,…,yk are definable in R and f=g1y1+⋯+gkyk.