Prolongation of integral domains
The motivation for this rather extensive and complicated paper is derived from the purely algebraic approach to the theory of overdetermined systems of partial differential equations. A system of (algebraic) differential equations represents a set of elements \(F_ 1,F_ 2,...,F_ n\) of a polynomial ring \(P=k[\delta^ py_ j: p\in {\mathbb{N}}^ m, j=1,...,n]\), where \(\delta_ 1,...,\delta_ n\) are derivation operators of a differential field k. The author suggests a general procedure for creating new equations of low order which are implied by the original one, i.e. he is dealing with inclusions \(k\subset A_ 0\subset...\subset A_ r\), where \(A_ s\) is an integral domain \(P_ s/{\mathfrak p}_ s\) \((P_ s=k[\delta^ py_ j: p\in {\mathbb{N}}^ m, | p| \leq s])\) with \({\mathfrak p}_ s\) a minimal prime component of \((F_ 1,....,F_ n)\cap P_ s\) (approximitely) and together with an action of a natural set of derivations of \(A_{r-1}\) into \(A_ r\) such that \(\delta_ i(A_{s-1})\subset A_ s\), \(0\leq s\leq r\). The main problem is whether there exists a differential integral domain \(A\supset A_ r\) whose derivations induce the already given actions \(\delta_ 1,...,\delta_ m\) (i.e. \(A_ r\) is ''embeddable'') and if it does not, to ''prolong'' the given system until r cannot be increased without changing \(A_ r.\) To solve the problem the author deals with a general situation described by ''differential kernels'' as any ring morphisms \(U\to V\) together with an action of \(\delta_ 1,...,\delta_ m\) of derivations of U into V. Using this notion he proves several theorems which solve in some aspect the problem presented here. Namely, he is mostly interested in complex conditions of existence of prolongations of differential kernels. Since the methods used in this paper are not very explicit he shows some explicit examples.
overdetermined systems of partial differential equations
derivation operators of a differential field
scientific article; zbMATH DE number 4004335