next | previous | forward | backward | up | top | index | toc | Macaulay2 website
TSpreadIdeals > tStronglyStableSeg

tStronglyStableSeg -- give the t-strongly stable segment with the given extremes

Synopsis

Description

the function tStronglyStableSeg(v,u,t) gives the set of t-spread monomials belonging to the strongly stable set generated by u and smaller than v, that is, $B_t[v,u]=\{w\in B_t\{u\}\ :\ v\geq_\mathrm{slex} w\}.$
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 set of monomials of $M_{n,d,t}$ 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 $x_i(u/x_j)\in N$.

Examples:

i1 : S=QQ[x_1..x_14]

o1 = S

o1 : PolynomialRing
i2 : tStronglyStableSeg(x_2*x_5*x_9*x_12,x_2*x_6*x_10*x_13,2)

o2 = {x x x x  , x x x x  , x x x  x  , x x x  x  , x x x x  , x x x x  ,
       2 5 9 12   2 5 9 13   2 5 10 12   2 5 10 13   2 6 8 10   2 6 8 11 
     ------------------------------------------------------------------------
     x x x x  , x x x x  , x x x x  , x x x x  , x x x x  , x x x  x  ,
      2 6 8 12   2 6 8 13   2 6 9 11   2 6 9 12   2 6 9 13   2 6 10 12 
     ------------------------------------------------------------------------
     x x x  x  }
      2 6 10 13

o2 : List
i3 : tStronglyStableSeg(x_2*x_5*x_9*x_12,x_2*x_6*x_10*x_13,3)

o3 = {x x x x  , x x x x  , x x x  x  , x x x x  , x x x x  , x x x  x  }
       2 5 9 12   2 5 9 13   2 5 10 13   2 6 9 12   2 6 9 13   2 6 10 13

o3 : List

See also

Ways to use tStronglyStableSeg :

For the programmer

The object tStronglyStableSeg is a method function.