At the time of the writing of this book, there is still only one approach to the numerical computation of a basis for the ideal defined by a polynomial system, for which software is commonly available: It is the computation of a Groebner basis, for a specified term order, by rational computation. This fact is presumably the main reason why polynomial algebra has not been widely used in Scientific Computing so far; cf. also section 1.5. In the approach to polynomial algebra taken in this book, Groebner bases have played a minor role; therefore, we will only briefly describe some principal aspects of their computation in section 10.1.

We will then turn our attention to regular systems of polynomials and normal set representations for their ideals. In describing a natural approach to their computation in section 10.2, we are aware of the fact that some aspects of such algorithms are not yet fully understood; we hope that our presentation will stimulate further research on this topic. Finally, in section 10.3, we will consider the truly numerical aspects of basis computation: The use of floating-point arithmetic and the application to empirical systems.