Magma allows one to homogenize a polynomial ring or ideal by use
of the Homogenization function, and also to restrict again to
the original ring with elimination automatically.
Homogenization(I, b) : RngMPol, RngIntElt, BoolElt -> RngMPol, Map
Given an ideal I of the polynomial ring P = R[x_1, ..., x_n], create a polynomial ring H as a single variable extension of P, the homogenized ideal J of H corresponding to I, and the homogenization map f: P -> H, and return J and f. If the argument b (standing for "before") is true, the homogenization variable is inserted before the current variables of P, so H is defined to be R[h, x_1, ..., x_n] and f maps P.i to H.(k + i) (so the x_i variables of P are mapped to the x_i variables of H). If the argument b is false, the homogenization variable is inserted after the current variables of P, so H is defined to be R[x_1, ..., x_n, h] and f maps P.i to H.i (so the x_i variables of P are mapped to the x_i variables of H). If the argument b is omitted, it is taken to be false, so the homogenization variable is introduced before. If the argument order is given, then H is constructed with the specified order; otherwise, the grevlex order is used for H by default. See the section on monomial orders for the valid values for the argument order. The image under f of a polynomial of P is the homogenization of f in H, while the image under f of an ideal of P is the homogenization ideal I^h in H. The inverse image under f of a polynomial of H is the restriction back to P (obtained by setting the homogenization variable to 1), while the inverse image under f of an ideal J of H is the restriction back to P of the elimination ideal J intersect K[x_1, ..., x_n].[Next] [Prev] [Right] [Left] [Up] [Index] [Root]