Chrastinová, Veronika; Tryhuk, Václav On the internal approach to differential equations. I: The involutiveness and standard basis. (English) Zbl 1399.58001 Math. Slovaca 66, No. 4, 999-1018 (2016). Summary: The article treats the geometrical theory of partial differential equations in the absolute sense, i.e., without any additional structures and especially without any preferred choice of independent and dependent variables. The equations are subject to arbitrary transformations of variables in the widest possible sense. In this preparatory Part 1, the involutivity and the related standard bases are investigated as a technical tool within the framework of commutative algebra. The particular case of ordinary differential equations is briefly mentioned in order to demonstrate the strength of this approach in the study of the structure, symmetries and constrained variational integrals under the simplifying condition of one independent variable. In full generality, these topics will be investigated in subsequent parts of this article. Cited in 2 ReviewsCited in 3 Documents MSC: 58A17 Pfaffian systems 35A30 Geometric theory, characteristics, transformations in context of PDEs 58E30 Variational principles in infinite-dimensional spaces Keywords:higher-order symmetries; diffiety; involutivity; standard basis PDF BibTeX XML Cite \textit{V. Chrastinová} and \textit{V. Tryhuk}, Math. Slovaca 66, No. 4, 999--1018 (2016; Zbl 1399.58001) Full Text: DOI arXiv OpenURL References: This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.