It has already been a long journey. From the snowy hills down to the valleys and up again, we have encountered many milestones
and discovered impressive tools. Meanwhile time has passed, the colours of the landscape have changed, and now it is time
to go back to the hills and do the harvesting.After we explain the main theorem for computing elimination modules in Section
3.4, many other ripening fruits will become ready for picking. Using the method of tag variables, we obtain new ways to perform
the basic operations on modules. Then, in Section 3.5, we learn how to compute saturations and how to check radical membership.
Other important applications are collected in Section 3.6, where we discuss ring homomorphisms. Among other things, we show
how to find presentations for the kernel and the image of a homomorphism of finitely generated algebras, how to solve the
implicitization problem, how to compute minimal polynomials of elements in a fine algebras, how to check membership in finitely
generated subalgebras, and how to analyze surjective and bijective homomorphisms between polynomial rings.
The final Section 3.7 of this chapter, and hence of this volume, represents the ultimate act of harvesting. It is devoted
to the problem of solving systems of polynomial equations effectively. At that stage of your reading, you will be challenged
to recall almost all the knowledge that you gathered during the journey, to become aware of the skills and the tools which
you have learned, and to use them to dig out the roots of systems of equations. Our last field of investigation also contains algorithms for checking whether the set of solutions
of a system of equations is finite, for computing squarefree parts of polynomials, and for finding radicals of zero-dimensional
ideals.