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Maurice Janet’s algorithms on systems of linear partial differential equations. (English) Zbl 1467.01007

The two authors, who are mathematicians located in Lyon, describe the emergence of formal methods in the theory of partial differential equations in the French school of mathematics through Janet’s work in the period 1913–1930. This work was mostly algebraic.
“As we explain in the following sections, the completion methods constitute the essential part of the theory developed by Janet. He introduced a procedure to compute a family of generators of an ideal having the involutive property, and called involutive bases in the modern language. This property is used to obtain a normal form of a linear partial differential equation system.” (p. 48)
Maurice Janet (1888–1983) was a mathematician born in Grenoble who, after stops in Nancy, Rennes, and Caen, finally became professor at the Sorbonne in 1945. He was president of the French Mathematical Society in 1948.
Janet had early contacts with German mathematics from his one semester study in Göttingen in late 1912 shortly before WWI. He kept a diary there which was edited by Laurent Mazliak [Le carnet de voyage de Maurice Janet à Göttingen. Collection “Essais”. Éditions Matériologiques (2013)] and is mentioned by the authors. The diary is mostly about Janet’s and others’ political positions but contains also interesting remarks about the foundations of mathematics. Although Janet uses Hilbert’s results on algebraic forms [JFM 22.0133.01] in his work on systems of partial differential equations, no personal contact with Hilbert is documented in the diary.
The authors put Janet’s work in the historical context of Pfaff’s problem, the Cauchy-Kowalewsky theorem and Grassmann’s differential rule, the latter being influential through the so-called Cartan-Kähler theory. The authors stress the role of the French mathematician Charles Riquier (1910) as a link to previous results. The authors quote at length from Janet’s French monograph on systems of partial differential equations [Leçons sur les systèmes d’équations aux dérivées partielles. Paris: Gauthier-Villars (1929; JFM 55.0276.01)]. They stress that his terminology is not the modern one, because he did not use the abstract results and methods from Emmy Noether’s theory of ideals from the early 1920s. Also, the work by the American L. E. Dickson in number theory (1913) and by the Russian N. M. Günther on algebraic forms (1913) was not immediately noticed by Janet, partly due to communication problems resulting from WWII. The lack of reference to the Russian work was criticized by J. Tamarkin’s review of Janet’s monograph in the Bulletin of the AMS in 1931. The authors also follow the impact and generalization of Janet’s work in mostly algebraic theories by W. Gröbner [Abh. Math. Semin. Univ. Hamb. 12, 127–132 (1938; Zbl 0018.30802)], H. Hironaka [Ann. Math. (2) 79, 205–326 (1964; Zbl 1420.14031)], and B. Buchberger [Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Innsbruck: Univ. Innsbruck, Mathematisches Institut (Diss.) (1965; Zbl 1245.13020)].

MSC:

01A60 History of mathematics in the 20th century
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
35-03 History of partial differential equations
35A25 Other special methods applied to PDEs

Biographic References:

Janet, Maurice
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References:

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