### Refine

Primary decomposition of an ideal in a polynomial ring over a field belongs to the indispensable theoretical tools in commutative algebra and algebraic geometry. Geometrically it corresponds to the decomposition of an affine variety into irreducible components and is, therefore, also an important geometric concept.The decomposition of a variety into irreducible components is, however, slightly weaker than the full primary decomposition, since the irreducible components correspond only to the minimal primes of the ideal of the variety, which is a radical ideal. The embedded components, although invisible in the decomposition of the variety itself, are, however, responsible for many geometric properties, in particular, if we deform the variety slightly. Therefore, they cannot be neglected and the knowledge of the full primary decomposition is important also in a geometric context.In contrast to the theoretical importance, one can find in mathematical papers only very few concrete examples of non-trivial primary decompositions because carrying out such a decomposition by hand is almost impossible. This experience corresponds to the fact that providing efficient algorithms for primary decomposition of an ideal I ae K[x1; : : : ; xn], K a field, is also a difficult task and still one of the big challenges for computational algebra and computational algebraic geometry.All known algorithms require Gr"obner bases respectively characteristic sets and multivariate polynomial factorization over some (algebraic or transcendental) extension of the given field K. The first practical algorithm for computing the minimal associated primes is based on characteristic sets and the Ritt-Wu process ([R1], [R2], [Wu], [W]), the first practical and general primary decomposition algorithm was given by Gianni, Trager and Zacharias [GTZ]. New ideas from homological algebra were introduced by Eisenbud, Huneke and Vasconcelos in [EHV]. Recently, Shimoyama and Yokoyama [SY] provided a new algorithm, using Gr"obner bases, to obtain the primary decompositon from the given minimal associated primes.In the present paper we present all four approaches together with some improvements and with detailed comparisons, based upon an analysis of 34 examples using the computer algebra system SINGULAR [GPS]. Since primary decomposition is a fairly complicated task, it is, therefore, best explained by dividing it into several subtasks, in particular, while sometimes only one of these subtasks is needed in practice. The paper is organized in such a way that we consider the subtasks separately and present the different approaches of the above-mentioned authors, with several tricks and improvements incorporated. Some of these improvements and the combination of certain steps from the different algorithms are essential for improving the practical performance.

The notion of Q-Gorenstein smoothings has been introduced by Kollar. ([KoJ], 6.2.3). This notion is essential for formulating Kollar's conjectures on smoothing components for rational surface singularities. He conjectures, loosely speaking, that every smoothing of a rational surface singularity can be obtained by blowing down a deformation of a partial resolution, this partial resolution having the property (among others) that the singularities occuring on it all have qG-smoothings. (For more details and precise statements see [Ko], ch. 6.). It is therefore of interest to construct singularities having qG-smoothings.