Finite differential inequality characterization for semialgebraic systems

Fix $m\ge 0$. Let $(A_{ij}(x))_{1\le i\le N,1\le j\le M}$ be a matrix of semialgebraic functions on $\mathbb{R}^n$, and let the system

involve unknowns $F=(F_1,\ldots,F_M)$ and data $f=(f_1,\ldots,f_N)$. For integers $\mu$ and $\nu$, consider linear \partial differential operators acting on $f\in C^\infty(\mathbb{R}^n,\mathbb{R}^N)$ of the forms

L_{\mu,\nu}f(x)=\sum_{i=1}^N\sum_{|\alpha|\le \bar m}a_{\mu\nu i\alpha}(x)\partial^\alpha f_i(x),

$$and$$

L'{\mu,\nu}f(x)=\sum{i=1}^N\sum_{|\alpha|\le \bar m}a'_{\mu\nu i\alpha}(x)\partial^\alpha f_i(x),

where the coefficients are semialgebraic and $\bar m\ge m$. **Finite differential inequality characterization.** There exist finitely many such operators, indexed by $1\le\mu\le\mu_{\max}$, with indices $1\le\nu\le\nu_\mu$ and $1\le\nu\le\nu'_{\mu}$, such that for every $f\in C^\infty(\mathbb{R}^n,\mathbb{R}^N)$, the system

admits a solution $F\in C^m(\mathbb{R}^n,\mathbb{R}^M)$ if and only if there exists $1\le\mu\le\mu_{\max}$ such that $L_{\mu,\nu}f\ge0$ on $\mathbb{R}^n$ for every $1\le\nu\le\nu_\mu$ and $L'_{\mu,\nu}f>0$ on $\mathbb{R}^n$ for every $1\le\nu\le\nu'_{\mu}$. The conjecture seeks a finite, effectively checkable differential characterization of solvability for these systems of inequalities; the supplied evidence says that the case $n=1$ remains open.