Multivariate polynomial division using // operator

[Jaya Kediya]

Hi, I'm trying to perform multivariate polynomial division where remainder is either 0 or a linear combination of leading terms of the divisors. One such example is shown in the picture below

 
I have tried doing the same with the // operator by putting divisors in a matrix but that seems to give a different output as shown

CODE :
R=QQ[x,y,z,MonomialOrder=>Lex];
g1=x*y-1;
g2=y^2-1;
g3=x^2*y+y^2*x+y^2;
I= ideal((g2),(g1))

(g3)//gens I

g3% gens I

OUTPUT:

      

Please help me spot the problem here thank you!

[Doug Torrance]

There's no problem with the answer, as it's also correct:

i8 : A = g3 // gens I

o8 = {2} | -x2y-2xy2+2y+1 |
     {2} | xy2+2y3        |

             2      1
o8 : Matrix R  <-- R

i9 : f = g3 % gens I

o9 = 2y + 1

o9 : R

i10 : gens I * A + f

o10 = | x2y+xy2+y2 |

              1      1
o10 : Matrix R  <-- R

i11 : oo_(0, 0) == g3

o11 = true

The problem is that, in general, multivariate polynomial division does not give a unique answer.  (That's one of the reasons for Groebner bases!)  In this example, g1 and g2 do *not* form a Groebner basis for I, so it's not surprising that we might get different results if we use slightly different algorithms (e.g., changing the order of the generators of the ideal).

This looks like one of the multivariate polynomial division examples from Cox, Little, and O'Shea, and here's some code implementing their division algorithm from Section 2.3:

polynomialDivision = (f, F) -> (
    s := #F;
    q := new MutableList from s:0;
    r := 0;
    p := f;
    while p != 0 do (
i := 0;
divisionOccurred := false;
while i < s and not divisionOccurred do (
    if leadTerm p % leadTerm F#i == 0 then (
q#i = q#i + leadTerm p // leadTerm F#i;
p = p - (leadTerm p // leadTerm F#i) * F#i;
divisionOccurred = true)
    else i = i + 1);
if not divisionOccurred then (
    r = r + leadTerm p;
    p = p - leadTerm p));
    (toList q, r))

This gives the same answer as in the book:

i14 : polynomialDivision(g3, {g2, g1})

o14 = ({x + 1, x}, 2x + 1)

[Jaya Kediya]

This is very helpful, thank you!! 

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[Adam Boocher]

Dear Doug, thank you very much for this!  I'm helping my students learn M2 and they are reading CLO.  This provides a great way to help them do Exercise 2.3.3 without having to code the algorithm by hand.