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4.8.7 Predefined Term-Orderings

The predefined term-orderings are:

* degree reverse lexicographic: DegRevLex (the default ordering)

* degree lexicographic: DegLex

* pure lexicographic: Lex

* pure xel: Xel

* elimination term-ordering: Elim(X:INDETS)

The first two term-orderings use the weights of the indeterminates for computing the degree of a monomial. If the indeterminates are given in the order x_1, ...,x_n, then x_1 > ... > x_n with respect to Lex, but x_1 < ... < x_n with respect to Xel.

In the last ordering, X specifies the variables that are to be eliminated. It may be a single indeterminate or a range of indeterminates. However, X may not be an arbitrary list of indeterminates; for that, see the command Elim (as opposed to the modifier Elim being discussed here). A range of indeterminates can be specified using the syntax < first-indet >..< last-indet >. Another shortcut: if there are indexed variables of the form, say, x[i,j], then Elim(x) specifies a term-ordering for eliminating all of the x[i,j].

Example

Use R ::= QQ[x,y,z], Lex; x+y+z; x + y + z ------------------------------- Use R ::= QQ[x,y,z], Xel; x+y+z; z + y + x ------------------------------- Use R ::= QQ[t,x,y,z], Elim(t); I := Ideal(t-x,t-y^2,t^2-xz^3); GBasis(I); [t - x, -y^2 + x, xz^3 - x^2] ------------------------------- Use R ::= QQ[x[1..5],y,z], Elim(x); -- term-ordering for eliminating all -- of the x[i,j]'s Ord(); Mat([ [1, 1, 1, 1, 1, 0, 0], [0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, -1, 0, 0], [0, 0, 0, -1, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0] ]) -------------------------------