# lexIdeal -- Compute the saturated lexicographic ideal in the given ambient space with given Hilbert polynomial

## Synopsis

• Usage:
lexIdeal (hp ,S)
lexIdeal (hp, n)
lexIdeal (d, S)
lexIdeal (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
• OrderVariables => ..., default value Down, Option to set the order of indexed variables
• Outputs:

## Description

Returns the saturated lexicographic ideal defining a subscheme of \mathbb{P}^{n} or Proj S with Hilbert polynomial hp or d.

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

## Ways to use lexIdeal :

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

## For the programmer

The object lexIdeal is .