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.
i5 : K = ZZ/32003 o5 = K o5 : QuotientRing |
i6 : R = K[z_0..z_5] o6 = R o6 : PolynomialRing |
i7 : M = random(ZZ^6,ZZ^150) o7 = | 3 9 2 3 7 3 9 0 0 1 5 2 4 8 9 9 8 2 0 4 2 7 0 5 6 5 6 8 1 1 2 2 9 3 5 | 6 3 6 5 9 1 6 9 5 7 4 6 9 4 3 3 4 6 4 8 8 9 1 8 9 0 2 1 0 7 1 9 7 8 2 | 3 7 0 6 4 8 6 8 1 5 0 1 7 2 2 6 8 6 3 3 7 2 7 9 9 4 7 3 5 4 5 8 2 4 5 | 6 6 2 3 5 9 2 3 8 6 1 1 4 7 2 9 5 5 1 1 1 5 0 4 6 9 5 9 5 9 9 1 2 2 3 | 8 9 6 5 0 1 6 7 2 7 4 4 6 9 6 3 6 8 8 8 3 7 2 3 4 1 1 3 7 0 3 5 3 4 0 | 6 6 9 7 4 2 4 9 2 4 4 5 4 0 8 4 5 5 7 8 0 3 7 0 8 4 8 4 7 5 8 4 9 5 6 ------------------------------------------------------------------------ 4 9 4 7 8 5 5 2 0 2 0 4 4 5 6 4 0 6 4 3 4 0 5 5 7 9 4 9 7 2 9 6 3 4 0 1 4 3 2 6 5 4 0 9 2 7 1 6 2 4 6 3 9 7 8 2 4 6 5 8 1 4 9 8 9 9 9 5 0 2 6 7 0 8 1 7 6 5 7 8 2 0 3 2 2 4 2 0 6 1 6 0 6 0 3 3 2 6 5 9 5 9 4 6 7 7 0 4 8 8 3 5 6 6 7 3 9 1 1 1 8 9 1 7 4 1 3 9 3 6 3 2 3 7 9 7 1 2 2 6 3 9 7 0 8 9 2 5 5 9 6 1 8 0 7 6 5 9 0 2 7 6 6 4 8 1 0 0 5 2 6 6 4 5 5 5 2 6 1 7 6 4 0 9 4 3 6 7 8 2 6 0 5 4 4 1 4 8 2 4 9 1 6 8 6 6 6 7 7 7 2 4 5 4 3 6 ------------------------------------------------------------------------ 1 2 6 0 2 6 5 6 0 8 3 7 0 1 5 5 1 9 2 8 0 0 1 4 0 9 3 5 2 9 6 4 4 0 0 9 3 1 0 0 0 7 5 2 1 3 7 3 6 3 4 9 0 3 9 9 8 4 6 7 6 6 1 2 0 9 8 7 2 8 9 1 2 2 6 0 6 3 0 9 0 0 2 9 6 8 2 6 4 6 5 0 3 4 6 5 6 2 3 5 6 8 4 0 9 3 6 6 1 9 4 7 0 2 8 1 7 2 5 1 9 9 4 8 8 1 0 8 9 3 5 9 8 1 2 8 0 0 3 8 0 3 9 1 8 2 8 4 7 9 5 1 8 4 8 1 3 6 5 2 3 2 2 7 2 3 7 7 1 3 1 4 8 1 6 5 6 3 1 8 0 5 9 4 3 0 3 1 7 2 4 8 7 3 5 1 9 2 5 6 2 6 9 3 7 4 1 7 1 0 7 2 3 1 0 3 ------------------------------------------------------------------------ 9 2 2 7 8 7 6 4 2 9 0 8 3 4 1 6 3 6 7 8 8 1 0 4 8 7 4 0 0 5 6 2 3 5 3 9 1 6 5 4 0 8 3 3 4 5 0 0 6 3 6 7 9 3 4 6 3 9 3 8 8 7 8 1 2 8 1 3 1 0 7 5 8 1 8 5 2 0 3 8 3 2 2 3 9 3 9 3 4 7 1 7 6 2 4 2 8 7 2 8 3 5 9 8 5 5 7 2 3 9 5 3 3 0 4 3 2 3 0 3 2 2 4 1 3 8 3 0 4 7 6 4 6 7 8 4 7 6 6 0 2 8 8 5 1 6 0 5 6 2 9 7 2 3 7 6 6 0 9 5 6 4 4 6 8 5 3 9 6 0 8 3 4 7 6 1 7 6 4 6 7 9 7 8 2 9 7 9 1 8 4 6 2 8 4 5 3 2 7 3 9 0 4 8 4 1 1 6 8 4 4 5 5 0 1 8 ------------------------------------------------------------------------ 8 8 5 8 6 6 4 | 4 0 5 0 5 0 7 | 6 7 3 9 4 9 0 | 1 3 2 8 2 0 5 | 1 5 0 2 0 7 1 | 8 4 9 6 8 0 9 | 6 150 o7 : Matrix ZZ <--- ZZ |
i8 : elapsedTime (inG,G) = projectivePoints(M,R); -- -0.487348 seconds elapsed |
i9 : elapsedTime H = projectivePointsByIntersection(M,R); -- 0.992491 seconds elapsed |
i10 : G == H o10 = true |
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.