## Abstract

Fix m ≥ 0, and let A = (Aij(x))_{1}≤i≤N,1≤j≤_{M} be a matrix of semialgebraic functions on R^{n} or on a compact subset E ⊂ R^{n}. Given f = (f1, . . ., fN) ∈ C^{∞}(R^{n}, 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}(R^{n}, R^{M} ) solution F = (F1, . . ., FM) if and only if f = (f1, . . ., fN) is annihilated by the linear partial differential operators.

Original language | English (US) |
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Pages (from-to) | 911-963 |

Number of pages | 53 |

Journal | Revista Matematica Iberoamericana |

Volume | 37 |

Issue number | 3 |

DOIs | |

State | Published - 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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