Involution I: Algebraic Theory

The first chapter was mainly concerned with the geometry behind the formal theory. For many problems a purely geometric approach is not sufficient (a concrete example is the question of proving the formal integrability of a differential equation) and additional algebraic ingredients are needed. These lead us to the title concept of this book: involution. Its algebraic theory will be the topic of the next four chapters. We will start in the current chapter with a more combinatorial approach leading to a special kind of Gröbner bases, the involutive bases; its algorithmic realisation is the topic of Chapter 4. In Chapter 5 we will show that the structure analysis of polynomial modules becomes significantly easier with the help of such bases. Finally, Chapter 6 will provide us with a homological explanation of the remarkable properties of (some) involutive bases. In Chapter 7 we will then return to differential equations and see how these algebraic ingredients appear there naturally in the analysis of the symbol of a differential equation.

In the first section we introduce the notion of an involutive division, a restriction of the usual divisibility relation of power products. As we are interested in applications beyond the classical commutative case, i. e. also in situations where the variables do not commute with each others, we introduce the theory for the monoid of multi indices and not for terms. Here we will already meet most of the key ideas about involution, as the later extension to polynomials will be fairly straightforward.