# groebnerFamily -- computes families of ideals with a specified initial ideal

## Synopsis

• Usage:
J = groebnerFamily M
J = groebnerFamily(M, L)
• Inputs:
• M, an ideal, a monomial ideal
• L, a list, a list of lists of standard monomials or smaller standard monomials for the generators of M
• Optional inputs:
• AllStandard => , default value false
• Weights => a list, default value null
• Variable => , default value t, or
• Outputs:
• F, an ideal, the groebner family, an ideal in the polynomial ring over the original variables and the parameters

## Description

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.