Abstract

Let m be a nonnegative integer, n a positive integer, N = f0; 1; 2; :::g and Nn = f1; : : : ; ng. A ranking is a total order of N m Nn such that (a; i) (b; j) implies (a + c; i) (b + c; j) for a, b, c 2 N m and i; j 2 Nn . We describe an approach to such rankings and a theorem which gives an explicit construction of an arbitrary ranking using nite real data. The case n = 1 corresponds to term-orderings of monomials which are crucial inputs for Buchberger's Grobner Basis algorithm for polynomial rings. The case n > 1 corresponds to rankings of partial derivatives which are inputs in algorithms in dierential algebra and Buchberger's algorithm for free modules over polynomial rings. A subclass of such rankings determined by nite integer data is given which is suÆcient for eective implementation of such rankings. This has been implemented in the symbolic language Maple. The rankings considered by Riquier are a special case of those considered here. Examples including appl...

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