# stronglyStableIdeals -- Compute the saturated strongly stable ideals in the given ambient space with given Hilbert polynomial

## Synopsis

• Usage:
stronglyStableIdeals (hp ,S)
stronglyStableIdeals (hp, n)
stronglyStableIdeals (d, S)
stronglyStableIdeals (d, n)
• Inputs:
• hp, , or
• hp, , a Hilbert polynomial;
• d, an integer, a positive integer corresponding to a constant Hilbert polynomial;
• S, , with numgens S > 1;
• n, an integer, number of variables of the polynomial ring.
• Optional inputs:
• CoefficientRing => ..., default value QQ, Option to set the ring of coefficients
• MaxRegularity => ..., default value null, Option to set the maximum regularity
• OrderVariables => ..., default value Down, Option to set the order of indexed variables
• Outputs:

## Description

Returns the list of all the saturated strongly stable ideals defining subschemes of \mathbb{P}^{n} or Proj S with Hilbert polynomial hp or d.

 i1 : QQ[t]; i2 : S = QQ[x,y,z,w]; i3 : stronglyStableIdeals(4*t, S) 5 4 2 2 4 5 2 o3 = {ideal (x, y , y z ), ideal (x*z, x*y, x , y z, y ), ideal (x*y, x , ------------------------------------------------------------------------ 2 4 2 3 x*z , y ), ideal (x*y, x , y )} o3 : List i4 : stronglyStableIdeals(4*t, 4) 5 4 2 2 4 5 o4 = {ideal (x , x , x x ), ideal (x x , x x , x , x x , x ), ideal (x x , 0 1 1 2 0 2 0 1 0 1 2 1 0 1 ------------------------------------------------------------------------ 2 2 4 2 3 x , x x , x ), ideal (x x , x , x )} 0 0 2 1 0 1 0 1 o4 : List i5 : hp = hilbertPolynomial(oo#0) o5 = - 4*P + 4*P 0 1 o5 : ProjectiveHilbertPolynomial i6 : stronglyStableIdeals(hp, S) 5 4 2 2 4 5 2 o6 = {ideal (x, y , y z ), ideal (x*z, x*y, x , y z, y ), ideal (x*y, x , ------------------------------------------------------------------------ 2 4 2 3 x*z , y ), ideal (x*y, x , y )} o6 : List i7 : stronglyStableIdeals(hp, 4) 5 4 2 2 4 5 o7 = {ideal (x , x , x x ), ideal (x x , x x , x , x x , x ), ideal (x x , 0 1 1 2 0 2 0 1 0 1 2 1 0 1 ------------------------------------------------------------------------ 2 2 4 2 3 x , x x , x ), ideal (x x , x , x )} 0 0 2 1 0 1 0 1 o7 : List i8 : stronglyStableIdeals(5, S) 5 2 4 2 3 2 o8 = {ideal (y, x, z ), ideal (x, y*z, y , z ), ideal (x, y , z , y*z ), ------------------------------------------------------------------------ 2 2 3 ideal (y*z, x*z, y , x*y, x , z )} o8 : List i9 : stronglyStableIdeals(5, 4) 5 2 4 2 3 2 o9 = {ideal (x , x , x ), ideal (x , x x , x , x ), ideal (x , x , x , x x ), 1 0 2 0 1 2 1 2 0 1 2 1 2 ------------------------------------------------------------------------ 2 2 3 ideal (x x , x x , x , x x , x , x )} 1 2 0 2 1 0 1 0 2 o9 : List

## Ways to use stronglyStableIdeals :

• "stronglyStableIdeals(ProjectiveHilbertPolynomial,PolynomialRing)"
• "stronglyStableIdeals(ProjectiveHilbertPolynomial,ZZ)"
• "stronglyStableIdeals(RingElement,PolynomialRing)"
• "stronglyStableIdeals(RingElement,ZZ)"
• "stronglyStableIdeals(ZZ,PolynomialRing)"
• "stronglyStableIdeals(ZZ,ZZ)"

## For the programmer

The object stronglyStableIdeals is .