Given a monomial ideal $M$ in a polynomial ring $R$, this computes the parameter families of homogeneous ideals where $M$ could be their initial ideal. These families are obtained from either the standard monomials to the generators of $M$, or the standard monomials smaller than the generators of $M$ but of the same degree as these generators. In the former case we obtain a family of all ideals where $M$ could be their initial ideal. In the latter case, we obtain such a family with respect to a given term order.
i1 : R = ZZ/32003[a,b,c,d]; |
i2 : M = ideal (a^2, a*b, b^2)
2 2
o2 = ideal (a , a*b, b )
o2 : Ideal of R
|
i3 : F = groebnerFamily M
2 2 2
o3 = ideal (a + t a*c + t b*c + t a*d + t b*d + t c + t c*d + t d , a*b +
1 2 4 5 3 6 7
------------------------------------------------------------------------
2 2 2
t a*c + t b*c + t a*d + t b*d + t c + t c*d + t d , b + t a*c +
8 9 11 12 10 13 14 15
------------------------------------------------------------------------
2 2
t b*c + t a*d + t b*d + t c + t c*d + t d )
16 18 19 17 20 21
ZZ
o3 : Ideal of -----[t , t ..t , t , t ..t , t , t ..t , t ..t , t ..t , t ..t , t ..t , t ..t , t ..t ][a..d]
32003 3 6 7 10 13 14 17 20 21 1 2 4 5 8 9 11 12 15 16 18 19
|
i4 : netList F_*
+---------------------------------------------------------------+
| 2 2 2 |
o4 = |a + t a*c + t b*c + t a*d + t b*d + t c + t c*d + t d |
| 1 2 4 5 3 6 7 |
+---------------------------------------------------------------+
| 2 2 |
|a*b + t a*c + t b*c + t a*d + t b*d + t c + t c*d + t d |
| 8 9 11 12 10 13 14 |
+---------------------------------------------------------------+
| 2 2 2|
|b + t a*c + t b*c + t a*d + t b*d + t c + t c*d + t d |
| 15 16 18 19 17 20 21 |
+---------------------------------------------------------------+
|
i5 : U = ring F o5 = U o5 : PolynomialRing |
i6 : T = coefficientRing U o6 = T o6 : PolynomialRing |
i7 : gens T
o7 = {t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t ,
3 6 7 10 13 14 17 20 21 1 2 4 5 8 9 11
------------------------------------------------------------------------
t , t , t , t , t }
12 15 16 18 19
o7 : List
|
i8 : gens U
o8 = {a, b, c, d}
o8 : List
|
Here, $F$ is the family of homogeneous ideals having $M$ as their initial ideal, under the term order of the ring of $M$.
The optional argument AllStandard is boolean, taking the value $true$ to compute the family of all homogeneous ideals with a given initial ideal and the value $false$ to compute the family with respect to a given order. The default value for this argument is false.
If $L$ is not given, then it is computed using standardMonomials (if AllStandard is true), or smallerMonomials (if AllStandard is false).
i9 : L = standardMonomials M
2 2 2 2
o9 = {{a*c, b*c, c , a*d, b*d, c*d, d }, {a*c, b*c, c , a*d, b*d, c*d, d },
------------------------------------------------------------------------
2 2
{a*c, b*c, c , a*d, b*d, c*d, d }}
o9 : List
|
i10 : F2 = groebnerFamily (M, L)
2 2 2
o10 = ideal (a + t a*c + t b*c + t a*d + t b*d + t c + t c*d + t d , a*b +
1 2 4 5 3 6 7
-----------------------------------------------------------------------
2 2 2
t a*c + t b*c + t a*d + t b*d + t c + t c*d + t d , b + t a*c +
8 9 11 12 10 13 14 15
-----------------------------------------------------------------------
2 2
t b*c + t a*d + t b*d + t c + t c*d + t d )
16 18 19 17 20 21
ZZ
o10 : Ideal of -----[t , t ..t , t , t ..t , t , t ..t , t ..t , t ..t , t ..t , t ..t , t ..t , t ..t ][a..d]
32003 3 6 7 10 13 14 17 20 21 1 2 4 5 8 9 11 12 15 16 18 19
|
Note that $F$ and $F_2$ are the same family, in this case.
The object groebnerFamily is a method function with options.