# projectivePoints -- produces the ideal and initial ideal from the coordinates of a finite set of projective points

## Synopsis

• Usage:
(inG,G) = projectivePoints(M,R)
• Inputs:
• M, , in which each column consists of the projective coordinates of a point
• R, a ring, homogeneous coordinate ring of the projective space containing the points
• Optional inputs:
• VerifyPoints => ..., default value true, Option to projectivePoints.
• Outputs:
• inG, an ideal, initial ideal of the set of projective points
• G, a list, list of generators for Grobner basis for ideal of projective points

## Description

This function uses a modified Buchberger-Moeller algorithm to compute a grobner basis for the ideal of a finite number of points in projective space.

 i1 : R = QQ[x_0..x_2] o1 = R o1 : PolynomialRing i2 : M = random(ZZ^3,ZZ^5) o2 = | 8 7 3 8 8 | | 1 8 7 5 5 | | 3 3 8 7 2 | 3 5 o2 : Matrix ZZ <--- ZZ i3 : (inG,G) = projectivePoints(M,R) 2 3 2 2 79285 204632 2 152667 o3 = (ideal (x , x , x x ), {x - -----x x - ------x + ------x x + 0 1 0 1 0 481 0 1 481 1 481 0 2 ------------------------------------------------------------------------ 817272 589744 2 3 318467 187373 2 77360 2 ------x x - ------x , x + ------x x x + ------x x - -----x x - 481 1 2 481 2 1 18648 0 1 2 4662 1 2 2331 0 2 ------------------------------------------------------------------------ 804017 2 294415 3 2 881693 273316 2 410665 2 ------x x + ------x , x x - ------x x x - ------x x + ------x x + 4662 1 2 2331 2 0 1 18648 0 1 2 2331 1 2 4662 0 2 ------------------------------------------------------------------------ 2169827 2 1563095 3 -------x x - -------x }) 4662 1 2 4662 2 o3 : Sequence i4 : monomialIdeal G == inG o4 = true

This algorithm may be faster than computing the intersection of the ideals of each projective point.

## Caveat

This function removes zero columns of M and duplicate columns giving rise to the same projective point (which prevent the algorithm from terminating). The user can bypass this step with the option VerifyPoints.