Smooth Solutions to Systems of Linear Inequalities on $\mathbb{R}$

Abstract

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.