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Primary decomposition: Algorithms and comparisons. (English) Zbl 0932.13019

Matzat, B. Heinrich (ed.) et al., Algorithmic algebra and number theory. Selected papers from a conference, Heidelberg, Germany, October 1997. Berlin: Springer. 187-220 (1999).
The primary decomposition of an ideal in a polynomial ring over a field is an important tool in commutative algebra and algebraic geometry. The search for efficient algorithms for computing primary decompositions is one of the big challenges of computational algebra. This article discusses and compares four such algorithms, including those found by D. Wang [“Characteristic sets and zero structures of polynomial sets,” RISC-Linz (Res. Inst. Symbolic Computation, Lect. Note, Univ. Linz 1989)], P. Gianni, B. Trager and G. Zacharias [J. Symb. Comput. 6, No. 2/3, 149-167 (1988; Zbl 0667.13008)], D. Eisenbud, C. Huneke and W. Vasconcelos [Invent. Math. 110, No. 2, 207-235 (1992; Zbl 0770.13018)], and T. Shimoyama and K. Yokoyama [J. Symb. Comput. 22, No. 3, 247-277 (1996; Zbl 0874.13022)].
The paper contains many subroutines in pseudo code, together with lots of examples. Most of the algorithms presented are implemented in SINGULAR. This paper represents an excellent and well-written overview of the different approaches to primary decomposition.
For the entire collection see [Zbl 0903.00035].

MSC:

13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13A15 Ideals and multiplicative ideal theory in commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
68W30 Symbolic computation and algebraic computation
13-04 Software, source code, etc. for problems pertaining to commutative algebra

Software:

SINGULAR