A fundamental problem in analysis and geometry is to decide when prescribed data admit a smooth solution subject to algebraic constraints. Let $A_{ij}$ be fixed semialgebraic coefficient functions, and let $f=(f_1,\ldots,f_N)$ be prescribed data. For systems of linear equations \[\sum_{j=1}^{M} A_{ij}(x)F_j(x)=f_i(x),\qquad i=1,\ldots,N,\] prior joint work with Fefferman gives a finite differential criterion: there exist finitely many linear differential operators $L_\nu$, depending only on the coefficients $A_{ij}$, such that \[\exists\, F=(F_1,\ldots,F_M)\in C^m\text{ solving the system}\quad \Longleftrightarrow \quad L_\nu f=0\quad \text{for all } \nu.\]
A natural question is whether an analogous finite criterion exists for systems of linear inequalities\[\sum_{j=1}^{M} A_{ij}(x)F_j(x)\le f_i(x),\qquad i=1,\ldots,N.\]
In several variables, $x\in \mathbb{R}^n$ with $n\ge 2$, the answer is negative.

This talk presents a positive resolution of the remaining one-dimensional case. We show that for systems of linear inequalities on $\mathbb{R}$ with fixed semialgebraic coefficients\[A_{ij}:\mathbb{R}\to \mathbb{R},\]the existence of a solution\[F=(F_1,\ldots,F_M)\in C^m(\mathbb{R},\mathbb{R}^M)\] satisfying\[\sum_{j=1}^{M} A_{ij}(x)F_j(x)\le f_i(x),\qquad x\in \mathbb{R},\quad i=1,\ldots,N,\] is characterized by finitely many linear ordinary differential inequalities in the data $f$. More precisely, there exist finitely many linear ordinary differential operators $L_\nu$, with semialgebraic coefficients, such that \[\exists\, F\in C^m(\mathbb{R},\mathbb{R}^M)\text{ solving the inequality system}\quad \Longleftrightarrow \quad L_\nu f(x)\ge 0\quad \text{for all } x\in \mathbb{R},\\nu=1,\ldots,K.\]
Interestingly, although finite differential criteria for inequalities fail in higher dimensions, they hold in dimension one.