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Term orderings on the polynomial ring. (English) Zbl 0584.13016
Computer algebra, EUROCAL ’85, Proc. Eur. Conf., Linz/Austria 1985, Vol. 2, Lect. Notes Comput. Sci. 204, 513-517 (1985).
[For the entire collection see Zbl 0568.00019.]
Let \(A=k[x_ 1,...,x_ n]\) be a polynomial ring over a field k and let T be the set of all monomials with coefficient 1 (terms) in A. Then a total order \(<\) on T is called a term ordering on A iff \(1<M\) for every \(M\in T-\{1\}\) and for any \(M_ 1\), \(M_ 2\) in T, \(M_ 1<M_ 2\) implies \(M_ 1\cdot M<M_ 2\cdot M\) for all M in T.
The main result of the paper states that term orderings on A can be classified by the following data: \((i)\quad an\) integer s with \(1\leq s\leq n\), \((ii)\quad a\) partition on n into s non-negative integers, \((iii)\quad an\) s-tuple of vectors in \({\mathbb{R}}^ n.\)
Term orderings are related to the so-called Gröbner bases of ideals in polynomial rings. Such bases are applied as a technical device which allows to perform algorithms for computations in a polynomial ring. These topics are developed by the author elsewhere, where also relations to commutative algebra are shown.
Reviewer: W.Bartol

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
06F05 Ordered semigroups and monoids
06F15 Ordered groups
68W30 Symbolic computation and algebraic computation
13-04 Software, source code, etc. for problems pertaining to commutative algebra