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fromCoordinates -- Compute the ideal of a point from its coordinates

Synopsis

• Usage:
I = fromCoordinates(L,C)
I = fromCoordinates(x,y,z, C)
• Inputs:
• L, a list, of three integers or field elements OR a list of lists of that type
• x, ,
• y, ,
• z, , integer or field coordinates of a point on C (assumed to be a plane curve)
• C, a ring, the ring in which the ideal of the point will be created
• Outputs:
• I, an ideal, of C, defining the subscheme corresponding to the list L of points.

Description

Convenient way to compute the ideal of a point on a plane curve, when the point is given by a list of its coordinates. If the coordinates are given as integers, they are interpreted as elements of the coefficient field The script returns an error if the point is not on the curve.

 i1 : S = ZZ/101[a,b,c] o1 = S o1 : PolynomialRing i2 : C = S/ideal"a3+b3-c3" o2 = C o2 : QuotientRing i3 : P = {0,1,1} o3 = {0, 1, 1} o3 : List i4 : Q = {1,1,0} o4 = {1, 1, 0} o4 : List i5 : fromCoordinates(P,C) o5 = ideal (a, - b + c) o5 : Ideal of C i6 : fromCoordinates({P,P},C) 2 2 2 o6 = ideal (a , - a*b + a*c, - a*b + a*c, b - 2b*c + c ) o6 : Ideal of C