Geodesic
On the conjecture of M.~Janet for systems of partial differential equations
Fundamentalʹnaâ i prikladnaâ matematika, Tome 25 (2024) no. 1, pp. 123-131.

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M. Janet in 1921 conjectured that an analytic solution to systems of $n$ consistent $m$-partial differential equations of $n$ unknown functions must contain at least one arbitrary function of $k$ variables, $k\geq m-1$. E. Kolchin at the Moscow International Congress in 1966 formulated an algebraic version of this conjecture. In the case of linear systems, it was proven by J. Johnson in 1978, but for nonlinear systems the question is still open. This paper shows that the generalized Janet conjecture does not hold for the intersection of $n$ differential hyperspaces in the case of any number of derivations $m>0$. 
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M. V. Kondratieva. On the conjecture of M.~Janet for systems of partial differential equations. Fundamentalʹnaâ i prikladnaâ matematika, Tome 25 (2024) no. 1, pp. 123-131. http://geodesic.mathdoc.fr/item/FPM_2024_25_1_a7/

[1] Kondrateva M., “Verkhnyaya granitsa minimiziruyuschikh koeffitsientov razmernostnogo mnogochlena Kolchina”, Programmirovanie, 36:2 (2010), 83–86 | MR | Zbl

[2] Kondrateva M. V., “Otsenka tipovoi differentsialnoi razmernosti sistemy lineinykh differentsialnykh uravnenii”, Fundament. i prikl. matem., 22:5 (2019), 259–269

[3] Goodearl K. R., “Global dimension of differential operator rings. II”, Trans. Amer. Math. Soc., 209:6 (1975), 5–85 | MR

[4] Janet M., “Sur les systémes aux dérivées partielles comprenant autant d'équations que de fonctions inconnues”, C. R. Acad. Sci. Paris, 172:1 (1921), 1637–1639 | MR

[5] Johnson J. L., “Systems of $n$ partial differential equations in $n$ unknown functions: the conjecture of M. Janet”, Trans. Amer. Math. Soc., 242 (1978), 329–334 | MR | Zbl

[6] Kolchin E. R., “Some problems in differential algebra”, Proc. Int. Congress of Mathematicians (Moscow, 1966), M., 1968, 269–276 | MR | Zbl

[7] Kolchin E. R., Differential Algebra and Algebraic Groups, Academic Press, 1973 | MR | Zbl

[8] Kondratieva M. V., Levin A. B., Mikhalev A. V., Pankratiev E. V., Differential and Difference Dimension Polynomials, Kluwer Academic, 1999 | MR | Zbl

[9] Ritt J., Differential Algebra, Amer. Math. Soc., New York, 1950 | MR | Zbl

[10] Sit W., “Typical differential dimension of the intersection of linear differential algebraic groups”, J. Algebra, 32:3 (1974), 476–487 | DOI | MR | Zbl