3.3.3 Term orderings

Any polynomial (resp. vector) in SINGULAR is ordered w.r.t. a term ordering (or, monomial ordering), which has to be specified together with the declaration of a ring. SINGULAR stores and displays a polynomial (resp. vector) w.r.t. this ordering, i.e., the greatest monomial (also called the leading monomial) is the first one appearing in the output polynomial, and the smallest monomial is the last one.

Remark: The novice user should generally use the ordering dp for computations in the polynomial ring ,resp. ds for computations in the localization .For more details, see Polynomial data.

In a ring declaration, SINGULAR offers the following orderings (but see also Monomial orderings):

1. Global orderings

   lp

   lexicographical ordering

   rp

   reverse lexicographical ordering, i.e. a lexicographical ordering from the right with 1 < x_1 < ... <x_n (should not be used as it reverses the "natural" x_1 > ... > x_n, reorder the variables instead)

   dp

   degree reverse lexicographical ordering

   Dp

   degree lexicographical ordering

   wp( intvec_expression )

   weighted reverse lexicographical ordering; the weight vector is expected to consist of positive integers only.

   Wp( intvec_expression )

   weighted lexicographical ordering; the weight vector is expected to consist of positive integers only.

   Global orderings are well-orderings, i.e., for each ring variable . They are denoted by a p as the second character in their name.

2. Local orderings

   ls

   negative lexicographical ordering

   rs

   negative reverse lexicographical ordering, i.e. a lexicographical ordering from the right (should not be used as it reverses the "natural" x_1 < ... < x_n, reorder the variables instead)

   ds

   negative degree reverse lexicographical ordering

   Ds

   negative degree lexicographical ordering

   ws( intvec_expression )

   (general) weighted reverse lexicographical ordering; the first element of the weight vector has to be non-zero.

   Ws( intvec_expression )

   (general) weighted lexicographical ordering; the first element of the weight vector has to be non-zero.

   Local orderings are not well-orderings. They are denoted by an s as the second character in their name.

3. Matrix orderings

   M( intmat_expression )

   intmat_expression has to be an invertible square matrix

   Using matrix orderings, SINGULAR can compute standard bases w.r.t. any monomial ordering which is compatible with the natural semi-group structure on the monomials. In practice, the predefined global and local orderings together with the block orderings should be sufficient in most cases. These orderings are faster than their corresponding matrix orderings since evaluation of a matrix ordering is more time consuming.

4. Extra weight vector

   a( intvec_expression )

   aa( intvec_expression )

   am( intvec_expression )

   an extra weight vector a( intvec_expression ), may precede any monomial ordering.
   aa allows larger degrees.
   am allows degrees for module generators (see example)

   Example for ordering am

   ring r=QQ,(x,y,z),(am(1,1,1,4,1,6),dp); r; ==> // coefficients: QQ ==> // number of vars : 3 ==> // block 1 : ordering am ==> // : names x y z ==> // : weights 1 1 1 ==> // : 3 module weights 4 1 6 ==> // block 2 : ordering dp ==> // : names x y z ==> // block 3 : ordering C [x,y,z]; ==> z*gen(3)+x*gen(1)+y*gen(2) deg([x]); ==> 5 deg([0,y]); ==> 2 deg([0,0,z]); ==> 7 deg([0,0,0,x]); ==> 1

5. Product ordering

   ( ordering [ ( int_expression ) ], ... )

   any of the above orderings and the extra weight vector may be combined to yield product or block orderings

   The orderings lp, dp, Dp, ls, ds, Ds and rp may be followed by an int_expression in parentheses giving the size of the block. For the last block the size is calculated automatically. For weighted orderings, the size of the block is given by the size of the weight vector. The same holds analogously for matrix orderings.

6. Module orderings

   ( ordering, ..., C )

   ( ordering, ..., c )

   sort polynomial vectors by the monomial ordering first, then by components

   ( C, ordering, ... )

   ( c, ordering, ... )

   sort polynomial vectors by components first, then by the monomial ordering

   Here a capital C sorts generators in ascending order, i.e., gen(1) < gen(2) < .... A small c sorts in descending order, i.e., gen(1) > gen(2) > .... It is not necessary to specify the module ordering explicitly since ( ordering, ..., C ) is the default.

   In fact, c or C may be specified anywhere in a product ordering specification, not only at its beginning or end. All monomial block orderings preceding the component ordering have higher precedence, all monomial block orderings following after it have lower precedence.

For a mathematical description of these orderings, see Polynomial data.