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, RN ), 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 Cm(Rn, RM ) solution F = (F1, . . ., FM) if and only if f = (f1, . . ., fN) is annihilated by the linear partial differential operators.
All Science Journal Classification (ASJC) codes
- Closures of ideals
- Linear system
- Polynomial system
- Real radical
- Semialgebraic partial differential operator
- Semialgebraic sets