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

We study the one-dimensional case of the problem of deciding when a system of linear inequalities \[
\sum_{j=1}^{M} A_{ij}(x)F_j(x) \leq f_i(x),
\qquad i=1,\ldots,N, \] with fixed semialgebraic coefficients $A_{ij}\colon \mathbb{R}\to\mathbb{R}$, admits a solution $F=(F_1,\ldots,F_M)\in C^m(\mathbb{R},\mathbb{R}^M)$. The analogous problem for systems of linear equations has a finite linear-differential criterion. For $x \in \mathbb{R}^{n \geq 2}$, the corresponding finite linear-differential-inequality criterion is known to fail. The obstruction comes from the fact that there are infinitely many directions to approach a point, and the fact that an infinite intersection of polytopes need not remain a polytope. The purpose of this paper is to prove that this obstruction disappears in dimension one. More precisely, solvability is characterized by finitely many linear ordinary differential inequalities in the data $f=(f_1,\ldots,f_N)$ with semialgebraic coefficients.