Homogenization of Ideals

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 performed automatically.

Homogenization(I, b): RngMPol, RngIntElt, BoolElt -> RngMPol, Map

Homogenization(I, b, order): RngMPol, RngIntElt, BoolElt, ... -> RngMPol, Map

Homogenization(I): RngMPol, RngIntElt, BoolElt -> RngMPol, Map

Homogenization(I, order): RngMPol, RngIntElt, BoolElt, ... -> RngMPol, Map

Given an ideal \(I\) of the polynomial ring \(P = R[x_1, \ldots, x_n]\), 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 \(f: P \rightarrow H\), 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 \(R[h, x_1, \ldots, x_n]\) 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 \(R[x_1, \ldots, x_n, h]\) 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 after the current variables of \(P\).

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 (Section Representation and 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 ideal obtained by setting the homogenization variable to 1.