General Methods for Solving Defining Systems

Abstract

In this chapter we describe a general approach to finding the solutions of defining systems. Since we are interested in classifying spherical tube hypersurfaces up to affine equivalence, we attempt to solve defining systems up to linear equiva- lence. More precisely, if F(x) is the solution of a defining system near the origin, we are interested in determining the hypersurface x ₀ =F(x) up to linear transformations in the variables x ₀, x. Thus, we begin the chapter by simplifying defining systems by means of such transformations. One of the consequences of our approach is the following globalization result: every spherical tube hypersurface in ℂⁿ⁺¹ extends to a spherical real-analytic hypersurface which is closed as a submanifold of ℂ^n+1.