Solutions to a system of equations for Cm functions

Charles Fefferman, Garving K. Luli

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish (US)
Pages (from-to)911-963
Number of pages53
JournalRevista Matematica Iberoamericana
Volume37
Issue number3
DOIs
StatePublished - 2021

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Keywords

  • Algorithms
  • Closures of ideals
  • Generators
  • Ideals
  • Linear system
  • Polynomial system
  • Real radical
  • Semialgebraic partial differential operator
  • Semialgebraic sets

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