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TSpreadIdeals > tStronglyStableIdeal

tStronglyStableIdeal -- give the smallest t-strongly stable ideal containing a given t-spread ideal

Synopsis

Description

the function tStronglyStableIdeal(I,t) gives the smallest t-strongly stable ideal containing I, that is, $B_t(G(I))$ where $G(I)$ is the minimal set of generators of $I$
We recall that if $u\in M_{n,d,t}\subset S=K[x_1,\ldots,x_n]$ then $B_t(u)$ is the smallest t-strongly stable ideal of $S$ containing $u.$
Moreover, a subset $N\subset M_{n,d,t}$ is called a t-strongly stable set if taking a t-spread monomial $u\in N$, for all $j\in \mathrm{supp}(u)$ and all $i,\ 1\leq i\leq j$, such that $x_i(u/x_j)$ is a t-spread monomial, then it follows that $x_i(u/x_j)\in N$.
A t-spread monomial ideal I is t-strongly stable if $[I_j]_t$ is a t-strongly stable set for all $j$, where $[I_j]_t$ is the t-spread part of the $j$-th graded component of I.

Examples:

i1 : S=QQ[x_1..x_9]

o1 = S

o1 : PolynomialRing
i2 : tStronglyStableIdeal(ideal {x_2*x_5*x_8},2)

o2 = ideal (x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x ,
             1 3 5   1 3 6   1 3 7   1 4 6   1 3 8   1 4 7   2 4 6   1 4 8 
     ------------------------------------------------------------------------
     x x x , x x x , x x x , x x x , x x x , x x x )
      1 5 7   2 4 7   1 5 8   2 4 8   2 5 7   2 5 8

o2 : Ideal of S
i3 : tStronglyStableIdeal(ideal {x_2*x_5*x_8},3)

o3 = ideal (x x x , x x x , x x x , x x x )
             1 4 7   1 4 8   1 5 8   2 5 8

o3 : Ideal of S

See also

Ways to use tStronglyStableIdeal :

For the programmer

The object tStronglyStableIdeal is a method function.