Smooth solutions to systems of linear inequalities on $\mathbb{R}$

Abstract

Fix an integer $m\geq 0$. We study the one-dimensional problem of deciding when a system of linear inequalities \begin{equation*} \sum_{j=1}^M A_{ij}(x)F_j(x)\leq f_i(x),\qquad i=1,\ldots,N, \end{equation*} with fixed semialgebraic coefficients $A_{ij}:\mathbb R\to\mathbb R$ and given functions \begin{equation*} f_i\in C^\infty(\mathbb R),\qquad i=1,\ldots,N, \end{equation*} admits a solution \begin{equation*} F=(F_1,\ldots,F_M)\in C^m(\mathbb R,\mathbb R^M). \end{equation*} The analogous problem for systems of linear equations admits a finite linear differential criterion, whereas for systems of inequalities such a criterion fails in general in dimensions at least two, already for continuous solutions. This paper gives a complete answer in one dimension for a system of linear inequalities. For $m=0,1,2$, the existence of a $C^m$ solution is characterized by a finite differential criterion. In contrast, for every $m\geq 3$, no finite criterion of this local differential type exists in general, even for a fixed constant-coefficient system. Thus, $m=2$ is the sharp regularity threshold for finite differential characterization of the one-dimensional inequality problem.