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Fundamentalnaya i Prikladnaya Matematika, 2023, Volume 24, Issue 3, Pages 129–138 (Mi fpm1938)  

The typical dimension of a system of first-order differential equations

M. V. Kondratieva

Moscow State University, Department of Mechanics and Mathematics, Leninskie Gory, Moscow, Russia, 119991
References:
Abstract: We prove that if a system of first-order partial differential equations in one variable has a nonzero Kolchin dimension polynomial, then its leading coefficient is equal to $1$. The notion of typical differential dimension plays an important role in differential algebra. Some of its estimations were proved by J. Ritt and E. Kolchin; they also advanced several conjectures that were later refuted. There are bounds for the typical differential dimension in codimension $1$ (E. Kolchin) and in the case of linear differential equations (D. Grigoriev). Note that in codimension $2$ for systems of linear differential equations in one indeterminate the well-known Bézout theorem holds, and in the case of several variables, we have earlier proved its analogue, which is not satisfied in higher codimensions. For nonlinear systems, in the general case, there are no exponential bounds yet (although it is known that the growth of the typical dimension is bounded by the Ackermann function).
Funding agency
This research has been supported by the Interdisciplinary Scientific and Educational School of Moscow University “Brain, Cognitive Systems, Artificial Intelligence.”
English version:
Journal of Mathematical Sciences (New York), 2024, Volume 283, Issue 6, Pages 912–918
DOI: https://doi.org/10.1007/s10958-024-07319-2
Document Type: Article
UDC: 512.628.2
Language: Russian
Citation: M. V. Kondratieva, “The typical dimension of a system of first-order differential equations”, Fundam. Prikl. Mat., 24:3 (2023), 129–138; J. Math. Sci., 283:6 (2024), 912–918
Citation in format AMSBIB
\Bibitem{Kon23}
\by M.~V.~Kondratieva
\paper The typical dimension of a~system of first-order differential equations
\jour Fundam. Prikl. Mat.
\yr 2023
\vol 24
\issue 3
\pages 129--138
\mathnet{http://mi.mathnet.ru/fpm1938}
\transl
\jour J. Math. Sci.
\yr 2024
\vol 283
\issue 6
\pages 912--918
\crossref{https://doi.org/10.1007/s10958-024-07319-2}
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    Фундаментальная и прикладная математика
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