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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, 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[x1, ..., xn],
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, x1, ..., xn] and f maps P.i to
H.(k + i) (so the xi variables of P are mapped to the xi
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[x1, ..., xn, h] and f maps P.i to
H.i (so the xi variables of P are mapped to the xi
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 Ih 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.
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