# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Gröbner bases: an algorithmic method in polynomial ideal theory. (English) Zbl 0587.13009
Multidimensional systems theory, Progress, directions and open problems, Math. Appl., D. Reidel Publ. Co. 16, 184-232 (1985).

[For the entire collection see Zbl 0562.00017.]

From the author’s introduction: ”Problems connected with ideals generated by finite sets $F$ of multivariate polynomials occur, as mathematical subproblems, in various branches of system theory,... The method of Gröbner bases is a technique that provides algorithmic solutions to a variety of such problems, for instance, exact solutions of $F$ viewed as a system of algebraic equations, computations in the residue class ring modulo the ideal generated by $F$, decision about various properties of the ideal generated by $F$, polynomial solution of the linear homogeneous equations with coefficients in $F$, word problems modulo ideals and in commutative semigroups (reversible Petri nets), bijective enumeration of all polynomial ideals over a given coefficient domain etc.”

The article under review gives a survey of the method of Gröbner bases and presents an improved version of their algorithmic construction. The author discusses applications as indicated above. One section deals with Gröbner bases for polynomial ideals over the integers, and the final section gives some information on implementations of the algorithm. While proofs are replaced by references to the original literature, many instructive examples illustrate the power of the algorithm.

Reviewer: Winfried Bruns

##### MSC:
 13F20 Polynomial rings and ideals 13P10 Gröbner bases; other bases for ideals and modules 13A15 Ideals; multiplicative ideal theory 68W30 Symbolic computation and algebraic computation