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tLexIdeal -- returns the t-spread lex ideal with a given ft-vector or with the same ft-vector of a given t-strongly stable ideal

Description

It has been proved that if I is a t-strongly stable ideal then a unique t-lex ideal with the same $f_\texttt{t}$-vector of I exists.
Let $\texttt{S}=K[x_1,\ldots,x_n]$, $\texttt{t}\geq 1$. The method tLexIdeal(S,f,t) gives the t-lex ideal of S with f as $f_\texttt{t}$-vector, if exists. The overloading method tLexIdeal(I,t) gives the t-lex ideal with the same $f_\texttt{t}$-vector of the t-strongly stable ideal I.

Examples:

i1 : S=QQ[x_1..x_8]

o1 = S

o1 : PolynomialRing
 
i2 : f={1,8,2,0,0}

o2 = {1, 8, 2, 0, 0}

o2 : List
 
i3 : I=tLexIdeal(S,f,2)

o3 = ideal (x x , x x , x x , x x , x x , x x , x x , x x , x x , x x , x x ,
             1 3   1 4   2 4   1 5   2 5   3 5   1 6   2 6   3 6   4 6   1 7 
     ------------------------------------------------------------------------
     x x , x x , x x , x x , x x , x x , x x , x x )
      2 7   3 7   4 7   5 7   1 8   2 8   3 8   4 8

o3 : Ideal of S
 
i4 : fTVector(I,2)==f

o4 = true
 
i5 : isTLexIdeal(I,2)

o5 = true
 
i6 : J=tStronglyStableIdeal(ideal {x_1*x_4*x_6},2)

o6 = ideal (x x x , x x x , x x x )
             1 3 5   1 3 6   1 4 6

o6 : Ideal of S
 
i7 : K=tLexIdeal(J,2)

o7 = ideal (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 8

o7 : Ideal of S
 
i8 : fTVector(J,2)==fTVector(K,2)

o8 = true

See also

Ways to use tLexIdeal:

  • tLexIdeal(Ideal,ZZ)
  • tLexIdeal(Ring,List,ZZ)

For the programmer

The object tLexIdeal is a method function.

The source of this document is in TSpreadIdeals.m2:1082:0.