On implicit systems of differential equations.
The author studies implicit autonomous systems of ordinary differential equations with polynomial dependency on both the unknown functions and their derivatives. This allows him to make heavy use of methods from commutative algebra and algebraic geometry in his analysis. After a brief review of the relevant methods, the main part of the article consists of an extensive discussion of the completion of such a system in ideal theoretic terms leading to explicit algorithms. The author also offers a geometric interpretation of completion in terms of the tangent variety of the constraint variety along the lines usually used in differential-geometric approaches to differential algebraic equations. Then he provides a field-theoretic formulation of the conditions under which initial data uniquely determine the initial derivatives and shows that this defines then in fact a birational equivalence. This result forms the basis for proving that systems satisfying these conditions possess (locally) unique analytic solutions. Finally, some concrete examples are discussed in detail.
scientific article; zbMATH DE number 2005882
scientific article