# pointsIdeal -- create an ideal of points

## Synopsis

• Usage:
pointsIdeal M
pointsIdeal(R, M)
• Inputs:
• M, , of size $m \times n$, either over the coefficient ring of $R$, or a polynomial ring $R$ with $m$ variables
• R, a ring, either the ring of $M$, or a polynomial ring with $m$ variables with coefficient ring the ring of $M$
• Outputs:
• an ideal, the homogeneous ideal in $R$ of the points which are the columns of $M$

## Description

In this example, we find the ideal of 6 general points in $\PP^3$. Since they are general, we can set the first 5 points to be in standard position (the coordinate points, and the point with all coordinates being 1).

 i1 : S = ZZ/32003[a..d] o1 = S o1 : PolynomialRing i2 : M = randomPoints(S, 6, Normalize => true) o2 = | 1 0 0 0 1 107 | | 0 1 0 0 1 4376 | | 0 0 1 0 1 -5570 | | 0 0 0 1 1 3187 | 4 6 o2 : Matrix S <--- S i3 : I = pointsIdeal M o3 = ideal (a*d + 13356b*d - 13357c*d, b*c + 9278b*d - 9279c*d, a*c - ------------------------------------------------------------------------ 15663b*d + 15662c*d, a*b - 4105b*d + 4104c*d) o3 : Ideal of S i4 : betti res I 0 1 2 3 o4 = total: 1 4 5 2 0: 1 . . . 1: . 4 2 . 2: . . 3 2 o4 : BettiTally