Solutions to a System of Equations for C^m Functions

2019/02/11 by Fefferman, Charles, Luli, Garving K.
#Classical Analysis and ODEs (math.CA) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1902.03691

Abstract

Fix m≥ 0, and let A=( A_ij( 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,⋯ ,f_N) ∈ C^∞ ( ℝⁿ,ℝ^N) , we consider the following system of equations ∑_j=1^MA_ij( x) F_j( x) =f_i( x) ( i=1,⋯ ,N) . 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,⋯, F_M) if and only if f=(f₁,⋯, f_N) is annihilated by the linear partial differential operators.

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