The Gröbner Cover by Antonio Montes

Author:Antonio Montes
Language: eng
Format: epub
ISBN: 9783030039042
Publisher: Springer International Publishing

(3)A polynomial system of equations , relating the tracer variables x and the auxiliary variables u, that describe the locus problem. The system F must contain m + 1 independent equations, in order to expect an n − 1-dimensional set of solutions in the x variables.

We shall consider the system F in the ring , taking the tracer variables as parameters and the remaining u as variables. Moreover, we take the mover variables w as the last n′ variables u, in order to make it easier to eliminate the remaining variables. It is also preferable to use order for the variables u as well as for the parameters x.

The simplest locus of this kind is a hypersurface, i.e., a simple polynomial equation in the tracer variables x = x 1…, x n of the n-dimensional space (m = 0), like a circle, a sphere or any polynomial equation. But the problems we are interested in are more general, and are derived from elaborate geometric constructions.

To deal with class A loci problems, we have to eliminate the variables u. For this objective, we can use the standard Gröbner lexicographical basis. But, as we shall see, we can obtain a more detailed description using the Gröbner Cover.

The Gröbner Cover is especially indicated for solving these types of problems. Moreover, the grobcov command gives the simplest canonical form of the discussion, and allow us to go into the problem in depth, define and determine a taxonomy of the locus components in an automatic way. The grobcov.lib library is to be used in the ring .

The question is to obtain the projection into the tracer space of the solution of the system F. This question can be formulated in the grobcov philosophy as follows: for which values of x do there exist solutions to F, and which ones are the corresponding values of the remaining variables.

Even if locus problems concern real solutions, in order to use automatic computations we will use the -Zariski topology of , as in the rest of the book. In practice we are interested in the projection of the locus L over the reals . This is not always straightforward and can cause problems in the computation. We discuss this question and give some partial solutions in the examples (see, for example, Sect. 7.5.3). This question leads to the following challenging problem: how to transform the Complex Gröbner Cover (GC) into the Real Gröbner Cover (RGC) . We shall refer to the complex locus and to the real locus in which we are properly interested. It is not sufficient to consider , because this can produce some undesired components that can be avoided if we use the real GC.

We have tried to elaborate on a didactical approach that includes the study of a large number of problems. In order to introduce the subject, let us consider a very simple problem where a tracer and a mover point clearly exist and has an elementary solution.

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