On Cm solutions to systems of linear inequalities

Abstract

Recent work of C. Fefferman and the first author [8] has demonstrated that the linear system of equations∑j=1MAij(x)Fj(x)=fi(x)(i=1,…,N), has a Cm solution F=(F1,…,FM) if and only if f1,…,fN satisfy a certain finite collection of partial differential equations. Here, the Aij are fixed semialgebraic functions.In this paper, we consider the analogous problem for systems of linear inequalities:∑j=1MAij(x)Fj(x)≤fi(x)(i=1,…,N). Our main result is a negative one, demonstrated by counterexample: the existence of a Cm solution F may not, in general, be determined via an analogous finite set of partial differential inequalities in f1,…,fN.