Lexicographical Order - Monomials

Monomials

In algebra it is traditional to order terms in a polynomial, by ordering the monomials in the indeterminates. This is fundamental, to have a normal form. Such matters are typically left implicit in discussion between humans, but must of course be dealt with exactly in computer algebra. In practice one has an alphabet of indeterminates X, Y, ... and orders all monomials formed from them by a variant of lexicographical order. For example if one decides to order the alphabet by

X < Y < ...

and also to look at higher terms first, that means ordering

... < X3 < X2 < X

and also

X < Yk for all k.

There is some flexibility in ordering monomials, and this can be exploited in Gröbner basis theory.