Solutions to a system of equations for C^m functions

Fix m ≥ 0, and let A = (Aij(x))₁≤i≤N,1≤j≤_M be a matrix of semialgebraic functions on Rⁿ or on a compact subset E ⊂ Rⁿ. Given f = (f1, . . ., fN) ∈ C^∞(Rⁿ, R^N ), we consider the following system of equations: M X Aij(x)Fj(x) = fi(x) for i = 1, . . ., N. j=1 In this paper, we give algorithms for computing a finite list of linear partial differential operators such that AF = f admits a C^m(Rⁿ, R^M ) solution F = (F1, . . ., FM) if and only if f = (f1, . . ., fN) is annihilated by the linear partial differential operators.