minimize -- turn a Gr\"obner basis computed using threaded Gr\"obner bases into a minimal one

Synopsis

Usage:

minimize(LineageTable)
Inputs:
- H, an instance of the type LineageTable, whose values are polynomials
Outputs:
- an instance of the type LineageTable, where values in H whose initial terms are divisible by others are replaced by null

Description

Scans values of a hash table H and retains only those whose initial terms are minimal generators of the ideal generated by the leading terms of the values of H. If the values of H constitute a Gr\"obner basis of the ideal they generate, this method returns a minimal Gr\"obner basis.

i1 : R = ZZ/101[a,b,c];

i2 : allowableThreads= 2;

i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")

                                   3
o3 = LineageTable{((0, 2), 0) => -c      }
                                   2
                  ((1, 2), 0) => -c
                             2
                  (0, 1) => a c
                               2
                  (0, 2) => b*c
                  (1, 2) => -a*c
                                2
                  0 => a*b*c + c
                          3       2
                  1 => - b c + a*b  + a*c
                        2
                  2 => b

o3 : LineageTable

i4 : minimize T

o4 = LineageTable{((0, 2), 0) => null}
                                  2
                  ((1, 2), 0) => c
                  (0, 1) => null
                  (0, 2) => null
                  (1, 2) => a*c
                  0 => null
                  1 => null
                        2
                  2 => b

o4 : LineageTable

Polynomials are normalized so that the leading coefficient is 1. Note that keys of non-minimal entries are retained, and the corresponding table value is null.

Ways to use minimize :

minimize(LineageTable)

For the programmer

The object minimize is a method function.