This chapter contains computer algebra methods tailored for commutative algebra. Some were specifically developed to study blowup algebras. Most however have a general nature and may be useful in other areas of commutative algebra.
The intrinsic intractability of very large scale computation in algebraic geometry and commutative algebra demand a continued refinement of techniques to ensure that the cost of computation be met largely by theoretical means. How to inform a computation of mathematical knowledge constitutes a most challenging problem.
The overall aim here is a discussion of basic constructions in and how they can effectively be carried out by symbolic computation programs.
The constructions are those for the radical of ideals, with some steps towards a facilitation of primary decomposition, the integral closure, primality testing, and the setting up of ideal transform computations. There is also a varied discussion of approaches to carry out primary decomposition of zero–dimensional ideals.
As a matter of general strategy, the constructions are mediated through the funnel of homological algebra. They can be characterized as successive layers of syzygies computations.
This has an advantage that often one can supply explicit formulas for the objects to be computed. On the other side of the ledger, the reader will not find here estimates for the complexity of the proposed constructions. There are several reasons for the omission, the main one being the near impossibility of estimating the complexity of Gröbner basis of ideals whose generators have many relationships.