On the internal approach to differential equations 1. The involutiveness\n and standard basis

2014/01/13 by Veronika Chrastinová, Chrastinová, Veronika, Václav Tryhuk +1
Mathematics · Physics and Astronomy · #13E05 #58A17 #58E99 #Algebraic and Geometric Analysis #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Waves and Solitons #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.1401.2764

openalex publication_date 2014/01/13 · openalex created_date 2025/10/27 · openalex updated_date 2026/07/28

Abstract

The article treats the geometrical theory of partial differential equations\nin the absolute sense, i.e., without any additional structures and especially\nwithout any preferred choice of independent and dependent variables. The\nequations are subject to arbitrary transformations of variables in the widest\npossible sense. In this preparatory Part 1, the involutivity and the related\nstandard bases are investigated as a technical tool within the framework of\ncommutative algebra. The particular case of ordinary differential equations is\nbriefly mentioned in order to demonstrate the strength of this approach in the\nstudy of the structure, symmetries and constrained variational integrals under\nthe simplifying condition of one independent variable. In full generality,\nthese topics will be investigated in subsequent Parts of this article.\n

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