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## 1.2 Examples for monomial orderings

Important orderings for applications are:

• The lexicographical ordering lp, given by the matrix:

resp. ls:

Remark 1..1   The positive lexicraphic ordering lp on is an elimanation ordering for

The definition of rings with these orderings in SINGULAR:
(Each line starting with // is a comment in SINGULAR.)

ring R1=0,(x(1..5)),lp;
ring R2=0,(x(1..5)),ls;


• The weighted degree reverse lexicographical ordering, given by the matrix

, (resp. ).

If wi = 1 (respectively wi = -1) for all i we obtain the degree reverse lexicographical ordering, dp (respectively ds).

The definition of rings with these orderings in SINGULAR:

ring R3=0,(x(1..5)),wp(2,3,4,5,6);
// correspond to w_i:2,3,4,5,6
ring R4=0,(x(1..5)),ws(2,3,4,5,6);
// correspond to w_i:-2,-3,-4,-5,-6
ring R5=0,(x(1..4)),dp;
ring R6=0,(x(1..4)),ds;


• An example for an elimination ordering for in is given by the matrix

with . In it is given by the same matrix with and .

The definition of a polynomial ring with an elimination ordering for x3 and x4 in SINGULAR:

ring E=0,(x(1..4)),(a(0,0,1,1),a(1,1),dp);
// correspond to w_i=1 for all i, r=2
// or simpler:
ring EE=0,(x(1..4)),(a(0,0,1,1),dp);


• The product ordering, given by the matrix

if the Ai define orderings on monomials given by the corresponding subsets of . Such an ordering can be used to compute in
• ( )
• ( )
• ( )
(See [GTZ], [GG], definition 2.1).

The definition of a ring with this ordering in SINGULAR:

ring P=0,(x(1..6)),(dp(4),ds(2));
// correspond to
// a first block of 4 variables with ordering dp
// and a second block of 2 variables with ordering ds


Next: 2. Standard bases Up: 1. Basic definitions Previous: 1.1 Monomial orderings
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