2022/02/11 by Marcello Malagutti, Malagutti, Marcello
Computer Science · Mathematics · Physics and Astronomy · #14P99 #15A54 #26B40 #26C99 #46E25 #Algebraic Geometry (math.AG) #FOS: Mathematics #Nonlinear Waves and Solitons #Numerical methods for differential equations #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2202.05815
openalex publication_date 2022/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Starting from the results of Charles Fefferman and Janos Kollár in \texitContinuous Solutions of Linear Equations [1], we adopt a new approach based on Fefferman's techniques of Glaeser refinement to show a more general result than the one proved by Kollár by using techniques from algebraic geometry. Considering a system of linear equations with semialgebraic (not only polynomial as in [1]) coefficients on ℝn, we get a necessary and sufficient condition for the existence of a continuous and semialgebraic solution on ℝn. This is different from what Fefferman and Luli obtained in Semialgebraic Sections Over the Plane since they stated their result for solutions of regularity Cm on the plane ℝ2. More in depth, we prove that a continuous and semialgebraic solution on ℝn exists if and only if there is a continuous solution i.e., if the Glaeser-stable bundle associated to the system has no empty fiber.