Polynomial Ideals and Their Varieties

As indicated in the Preface, solutions of the fundamental questions addressed in this book, the center and cyclicity problems, are expressed in terms of the sets of common zeros of collections of polynomials in the coefficients of the underlying family of systems of differential equations. These sets of common zeros are termed varieties. They are determined not so much by the specific polynomials themselves as by larger collections of polynomials, the so-called ideals that the original collections of polynomials generate. In the first section of this chapter we discuss these basic concepts: polynomials, varieties, and ideals. An ideal can have more than one set of generating polynomials, and a fundamental problem is that of deciding when two ideals, hence the varieties they determine, are the same, even though presented by different sets of generators. To address this and related isssues, in Sections 1.2 and 1.3 we introduce the concept of a Gröobner basis and certain fundamental techniques and algorithms of computational algebra for the study of polynomial ideals and their varieties. The last section is devoted to the decomposition of varieties into their simplest components and shows how this decomposition is connected to the structure of the generating ideals. For a fuller exposition of the concepts presented here, the reader may consult [1, 18, 23, 60].