ring result ring

  Interactively define a ring. You are asked for some obvious information, and then for some more obscure information.

The following information is requested:

1. Characteristic of the ring - a positive prime number less than about 32000 is required. If you want to simulate characteristic zero, simply type <return>. A large characteristic is then selected.
2. The number of variables - must be positive. There is also an upper limit on the number of variables (100 on Macintosh, 512 on Unix systems).
3. Names of the variables (see section 3.4 for the syntax of ring indeterminates). These should be given without intervening spaces. Sequences of variables may be generated by using ``-'' between two variables, e.g: ``a-f'' expands to ``abcdef'', and ``a[1]-b[2]'' expands to ``a[1]a[2]b[1]b[2]''. If not enough variables are given, you will be prompted for more. If too many are given, the extras are ignored (with a warning).
4. Weights of the variables. An integer list is expected. The weight of each variable must be a positive integer. If <return>\ is entered, then the variables are all assigned weight one. If too few weights are given, then the last weight given is assigned to the remaining variables.
5. The monomial order. Typing <return> selects the reverse lexicographic order. Typing in a number (which is less than the number of variables) selects a product order (see section 5.4 for details, and the general form expected for the input).

Very often, one types <return> to requests (1), (4) and (5). The following examples define rings, using these defaults.

%  ; first define a ring with variables {a,b,c,d}:
%  ring R
! characteristic (if not 31991)       ? <return>
! number of variables                 ?  4
!   4 variables, please               ?  a-d
! variable weights (if not all 1)     ? <return>
! monomial order (if not rev. lex.)   ? <return>
;   largest degree of a monomial        : 512 

% <return>
; second, define a ring with variables {x,y,z,a[1,1], ..., a[2,2]}
%  ; (7 variables in all)
%  ring S
! characteristic (if not 31991)       ? <return>
! number of variables                 ?  7
!   7 variables, please               ?  xyza[1,1]-a[2,2]
! variable weights (if not all 1)     ? <return>
! monomial order (if not rev. lex.)   ? <return>
;   largest degree of a monomial        : 76 

% <return>
; display these rings using the "pring" command
%  pring R
; ring R
; characteristic           : 31991
; number of variables      : 4
; variables                : abcd
; weights                  : 1 1 1 1 
; monomial order           : 4 c 
; top degree of a monomial : 512 

%  pring S
; ring S
; characteristic           : 31991
; number of variables      : 7
; variables                : xyza[1,1]a[1,2]a[2,1]a[2,2]
; weights                  : 1 1 1 1 1 1 1 
; monomial order           : 7 c 
; top degree of a monomial : 76

The standard (Gröbner) basis computations in Macaulay require every ideal to be homogeneous, and each matrix to be graded. The weights of the ring variables can often be set so that an ideal is homogeneous. For example, let . This ideal is not homogeneous. However, if we set the weight of s, t to one, and the weight of x to 3, y to 6, and z to 5, then each generator of I is homogeneous with respect to these weights. This is often used with elimination of variables.

%  ; define a ring with variables {a,b,c,d,e} having weights 5,3,1,1,1
%  ; notice that the last "1" typed is repeated, so that the variables
%  ; c,d,e all have weight one.
%  ring R
! characteristic (if not 31991)       ? <return>
! number of variables                 ?  5
!   5 variables, please               ?  a-e
! variable weights (if not all 1)     ?  5 3 1
! monomial order (if not rev. lex.)   ? <return>
;   largest degree of a monomial        : 372 

%  pring R
; ring R
; characteristic           : 31991
; number of variables      : 5
; variables                : abcde
; weights                  : 5 3 1 1 1 
; monomial order           : 5 c 
; top degree of a monomial : 372

See Appendix A for more examples.

Caveat: the weight of each variable must be a positive integer. If any of the elements of the integer list are negative, every weight is increased by the same amount in order for the weights to be all positive.

The format for monomial orders is rather complicated, and a whole section (section 5.4) is devoted to it.