@article{fc1a6c6c367b4fa88df78acbd8dcb0c2,
title = "Solutions to a system of equations for Cm functions",
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.",
keywords = "Algorithms, Closures of ideals, Generators, Ideals, Linear system, Polynomial system, Real radical, Semialgebraic partial differential operator, Semialgebraic sets",
author = "Charles Fefferman and Luli, {Garving K.}",
note = "Funding Information: Dr Joynt Maddox receives research support from the National Heart, Lung, and Blood Institute (R01HL143421) and National Institute on Aging (R01AG060935), and previously did contract work for the US Department of Health and Human Services. Dr Bleser has previously received consulting fees from Merck on vaccine litigation unrelated to this work and serves as Board Vice President (uncompensated) for Shepherd{\textquoteright}s Clinic, a clinic providing free healthcare to the uninsured in Baltimore, MD. Dr Desai works under contract with the Centers for Medicare and Medicaid Services to develop and maintain performance measures used for public reporting and pay for performance programs, and he reports consulting for Amgen, Boehringer Ingelheim, Cyto-kinetics, Medicines Company, Relypsa, Novartis, and SCPharmaceuticals. Dr O{\textquoteright}Brien reports research grants from BMS and Novartis. Dr Psotka has received consulting fees from Amgen, Cytokinetics, and Windtree and receives a research fellowship from the US Food and Drug Administration. Dr Wadhera receives research support from the National Heart, Lung, and Blood Institute (grant K23HL148525-1) at the National Institutes of Health. He previously served as a consultant for Regeneron, outside the submitted work. The other authors report no conflicts. Funding Information: The Value in Healthcare Initiative is a collaboration of the American Heart Association and the Robert J. Margolis, MD, Center for Health Policy at Duke University. The Value in Healthcare Initiative is supported in part by the American Heart Association and industry partners (Amgen, Bristol-Myers Squibb, the Bristol-Myers Squibb-Pfizer Alliance, and the Sanofi-Regeneron Alliance). Publisher Copyright: {\textcopyright} European Mathematical Society",
year = "2021",
doi = "10.4171/rmi/1217",
language = "American English",
volume = "37",
pages = "911--963",
journal = "Revista Matematica Iberoamericana",
issn = "0213-2230",
publisher = "Universidad Autonoma de Madrid",
number = "3",
}