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## Examples

Suppose we want to compute the Gröbner basis of the set

(see Section 2.1) for the following setup:

• Coefficient Domain: Rational Numbers
• Term Ordering: Total Degree Inverse Lexical
• Pair Ordering: Total Degree Inverse Lexical
• Tracing: Inform about setup, leading power product of non-zero normal forms, application of criteria, and zero reductions.

The dialog would look as follows:

```unix% groebner_setup
============================================
============================================
Coefficient Domain ................... C
Term Ordering ........................ TO
Pair Ordering ........................ PO
Normal Form Algorithm ................ N
List Operations ...................... L
Representation of Power Products ..... PoPr
Representation of Pure Polynomials ... PuPo
Polynomial Structure ................. PoSt
Representation of Polynomial Sets .... PoSe
Tracing .............................. T
EXIT ................................. X
============================================
=================================================
Current setting is: rational_numbers
=================================================
Rational Numbers ........ RN
Rational Functions ...... RF
Finite Field Zp ......... FF
Galois Field ............ GF
Floting Point Numbers ... FPN
Leave Unmodified ........ <return>

============================================
============================================
Coefficient Domain ................... C
Term Ordering ........................ TO
Pair Ordering ........................ PO
Normal Form Algorithm ................ N
List Operations ...................... L
Representation of Power Products ..... PoPr
Representation of Pure Polynomials ... PuPo
Polynomial Structure ................. PoSt
Representation of Polynomial Sets .... PoSe
Tracing .............................. T
EXIT ................................. X
============================================
=================================================
Current setting is: lexical
=================================================
Total Degree Inverse Lexical ... TDI
Total Degree Lexical ........... TD
Lexical ........................ Lex
Matrix ......................... Mat
Leave Unmodified ............... <return>

============================================
============================================
Coefficient Domain ................... C
Term Ordering ........................ TO
Pair Ordering ........................ PO
Normal Form Algorithm ................ N
List Operations ...................... L
Representation of Power Products ..... PoPr
Representation of Pure Polynomials ... PuPo
Polynomial Structure ................. PoSt
Representation of Polynomial Sets .... PoSe
Tracing .............................. T
EXIT ................................. X
============================================
=================================================
Current setting is: total_degree_inverse_lexical
=================================================
Total Degree Inverse Lexical ... TDI
Total Degree Lexical ........... TD
Lexical ........................ Lex
Matrix ......................... Mat
Leave Unmodified ............... <return>

============================================
============================================
Coefficient Domain ................... C
Term Ordering ........................ TO
Pair Ordering ........................ PO
Normal Form Algorithm ................ N
List Operations ...................... L
Representation of Power Products ..... PoPr
Representation of Pure Polynomials ... PuPo
Polynomial Structure ................. PoSt
Representation of Polynomial Sets .... PoSe
Tracing .............................. T
EXIT ................................. X
============================================

Group     1: Include IO-library header file:                            Y
Group     2: Print algorithm variant :                                  N
Group     4: Print the current setup (general):                         Y
Group     8: Print the current setup (details):                         Y
Group    16: Statistics:                                                N
Group    32: Inform about reduction:                                    N
Group   128:    Label (i.e. number) of this polynomial:                 N
Group   256:    Leading power product:                                  Y
Group   512:    Size (i.e. number of terms):                            N
Group  1024:    Whole polynomial:                                       N
Group  2048: Inform about application of criteria:                      Y
Group  4096: Announce zero-reduction:                                   Y
Group  8192: Print size of basis and pair set:                          N
Group 16384: Details during reduction:                                  N
============================================
============================================
Coefficient Domain ................... C
Term Ordering ........................ TO
Pair Ordering ........................ PO
Normal Form Algorithm ................ N
List Operations ...................... L
Representation of Power Products ..... PoPr
Representation of Pure Polynomials ... PuPo
Polynomial Structure ................. PoSt
Representation of Polynomial Sets .... PoSe
Tracing .............................. T
EXIT ................................. X
============================================
Setup finished
unix% groebner

<Compiling Messages>

Done.
Path for Input-file: /disk2/users/groebner/Groebner/input
Type input-file name ( <RETURN> = stdin, ? = help):
Path for Output-file: /disk2/users/groebner/Groebner/input
Output-file name ( <RETURN> = stdout, ? = help):
***********************************************************
Input File:  stdin
***********************************************************
***********************************************************
Output File: stdout
***********************************************************
============================
WELCOME TO GROEBNER

Version 2.1
I/O library Version 2.1
by the RISC-Linz Institute

Good Luck ...
============================
SACLIB initialized with 500000 cells in SPACE.

Compute GROEBNER BASIS (Crude Version) ........... 1
Compute GROEBNER BASIS (Criteria) ................ 2
Compute GROEBNER BASIS (Reduce All) .............. 3
Compute GROEBNER BASIS (Small Pair Set) .......... 4
EXIT ............................................. 0
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Current Compiled Version:
Groebner Library: Version 2.1
List Operations: Recursive
Coefficient Domain: Rational Numbers
Term Ordering: Total Degree Inverse Lexicographic Ordering
Pair Ordering: Total Degree Inverse Lexicographic Ordering
Normal Form Algorithm: Complete Normal Form
-----------------------------------------------------------
Node Type: hp9000s700
Host: hotblack
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Groebner Basis with Reduce All
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Initializing Global Variables for GROEBNER ...
Done.
Initializing Global Variables for GROEBNER-IO ...
Enter ring-variables in ascending order (? for help): z y x;
Done.
===========================================================
Number of Variables: 3
Ring Variables: x > y > z
===========================================================
Enter Polynomial Set (? for help):

{16*x^2 + 4*x*y^2 - 4*z + 1,
4*x + 2*y^2*z + 1,
2*x^2*z - x - 2*y^2}
{
2
2 y z + 4 x + 1,
2       2
2 x z - 2 y  - x,
2       2
4 x y  + 16 x  - 4 z + 1
}
New polynomial:
[0 2 1]
New polynomial:
[2 0 1]
New polynomial:
[1 2 0]
New polynomial:
[2 0 0]
New polynomial:
[0 0 3]
New polynomial:
[0 4 0]
Zero-reduction
Product Criterion
Zero-reduction
Product Criterion
Zero-reduction
Product Criterion
Product Criterion
Product Criterion
Groebner Basis =
{
2      2   1  2   3     1
x  - 2 y  + - z  - - x - - z,
2      4     8

3      2   3       1  2
z  + 2 y  - - x z - - z  + 9 x + 2,
2       4

2          1
y z + 2 x + -,
2

2      2      2         1     1
x y  + 8 y  - 2 z  + 3 x - - z + -,
2     4

4      2       2   1          2          5     9
y  - x z  - 32 y  + - x z + 8 z  - 12 x + - z - -
4                     2     8
}
User Time: 0.030000

Compute GROEBNER BASIS (Crude Version) ........... 1
Compute GROEBNER BASIS (Criteria) ................ 2
Compute GROEBNER BASIS (Reduce All) .............. 3
Compute GROEBNER BASIS (Small Pair Set) .......... 4
EXIT ............................................. 0