CoCoA » CoCoALib

This example was generated by simplifying the original big example found by L.Robbiano

smaller

W := mat([[0, 1, 1]]);
R := NewPolyRing(QQ, first(lettere, NumCols(W)), MakeTermOrdMat(W), 1);
use R;

LL := [
       a^4*b -a^4*c -2*a^3*c +a^2*b +2*a^2*c -3*a*c +c,
       -4*a^4*c^2 -a^3*c^2 +2*a^2*b*c +3*a^2*c^2 -6*a*c^2 +2*c^2,
       a^5*b*c -a^5*c^2 -2*a^4*c^2 -a^3*b*c
       ];

L := MinGens(ideal(LL));  indent(L);

L[2] IsIn ideal(L[1],L[3]);
GenRepr(L[2], ideal(L[1], L[3]));

SetVerbosityLevel(120);
--======================================================
W := mat([[0, 1, 1]]);
R := NewPolyRing(QQ, first(lettere, NumCols(W)), MakeTermOrdMat(W), 1);
use R;

LL := [  a*b +c,  a^2*c,  a^2*b ];
L := MinGens(ideal(LL));  indent(L);  --->  [a*b +c,  a^2*c,  -a*c]

L[2] IsIn ideal(L[1],L[3]);  ---> true

1. a*b +c
2. a^2*c
3. a^2*b ---> -a*c

When all weights are positive,
if deg(T1) = deg(T2) and T1 | T2 then T1 = T2
(thanks J.Abbott for pointing that out)
so T2+... could be reduced by T1+... as well as the opposite, making the order in which we deal with them irrelevant.

This is not the case here:
deg(a*c) = deg(a^2*c) and a*c | a^2*c
but a^2*c was the one found first and is not reduced by the new one.

Notice that we are following this ordering

 [[0, 1, 1],
  [1, 0, 0],
  [0, 0, -1]])

 a*b  <  a^2*c  <  a^2*b

Now I'm inclined to say there is no trivial way out.
The good point is that we could just interreduce the polynomials with the same degree, but that would require changing the "homogeneous" code ad-hoc and is not trivial.

I have inserted a "quick hack" solution: just call HasPositiveGrading inside myMinGens.
It detects the examples here. Checking in so that we can make an interim release tomorrow,

For the time being: should we give error? (as if non-homogeneous?)

Perhaps ERR::NYI ?

This is how the usual MinGens labelling can go horribly wrong with non-negative gradings:

S := NewPolyRingWeights(QQ, "x,y,a", mat([ [1, 1, 0] ]));
Use S;

G := [ y*a -y, x*a, x*a -y*a -x ];

This is an extract with verbosity = 200: (polys in GB are indexed from 0)

A new poly coming from a pair reduces the tails of the polinomial labelled as minimal (in the same degree!)
thus making them non-minimal --> [y*a, x*a, -x]

Note: the final GB is [x, y]

===============================
 [D2,L121,myReduceCurrentSPoly]  <InputPoly>
 [D2,L200,myReduceCurrentSPoly]   --before: y*a -y      --after:  y*a -y  --> GB[0]

 [D2,L121,myReduceCurrentSPoly]  <InputPoly>
 [D2,L200,myReduceCurrentSPoly]   --before: x*a         --after:  x*a  --> GB[1]

 [D2,L121,myReduceCurrentSPoly]  <InputPoly>
 [D2,L200,myReduceCurrentSPoly]   --before: x*a -y*a -x --after:  -x -y  --> GB[2]

---  all 3 are labelled "minimal generator"  ---

 [D2,L121,myReduceCurrentSPoly]  <1,2>
 [D2,L200,myReduceCurrentSPoly]   --before: -y*a   --after:  -y  --> GB[3]

---  -y comes from a pair, so not a "minimal generator" 
---  but is used to reduce the *tails* (interreduction)
---     so modifies y*a -y --> y*a (!!!) making it *not minimal*
---  in fact,  y*a -y  is a multiple of y (cannot happen with positive grading)

 [D2,L121,myReduceCurrentSPoly]  <0,3>
 [D2,L200,myReduceCurrentSPoly]   --before: 0   --after:  0

[D1,L100,myDoGBasis]  --Final clean up ...