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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

##### MSC:
 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