On the global existence of holomorphic solutions and the semi-global existence of real analytic solutions of linear partial differential equations
This paper contains an erroneous claim at a key-point in the reasoning: Theorem 3.5 is correct only in the two-dimensional case. The correct statement can be found by \textit{H. Suzuki} [Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 11, 253-258 (1972; Zbl 0255.35005)] for the first order equations, and, for higher order equations by the reviewer and \textit{Y. Takei} [Bicharacteristical convexity and the semi-global existence of holomorphic solutions of linear differential equations with holomorphic coefficients, RIMS-Preprint 581 (1987)]. Since another seemingly interesting part of this paper, i.e., {\S} 5, is based upon Theorem 3.5, it is not a mathematical claim. It should also be noted that, although the author uses the expression ``Theorem 3.5'', there is no proof for it. Apparently the author jumped to the erroneous conclusion that the bicharacteristical convexity entails the positive-definiteness of the generalized Levi form; simple and interesting counter-examples to this claim can be found in the above quoted papers. The reviewer hopes that the editorial board of Journal d'Analyse Mathématique announces for the benefit of the mathematical world that this paper contains serious errors.
scientific article; zbMATH DE number 4027973
scientific article