CoCoA:Homogenized

Homogenized

homogenize with respect to an indeterminate

Description

This function returns the homogenization of E with respect to the

indeterminate X, which must have weight 1. Note that in the case

where E is an ideal, Homogenized returns the ideal generated by the homogenizations of all the elements of E, not just the homogenization of the generators of E (see the example, below). The coefficient ring must be a field for this function to work reliably.

Example

  Use R ::= Q[x,y,z,w];
  Homogenized(w, x^3-y);
x^3 - yw^2
-------------------------------
  Homogenized(w, [x^3-y, x^4-z]);
[x^3 - yw^2, x^4 - zw^3]
-------------------------------
  I := Ideal(x^3-y, x^4-z);
-- same as  Homogenized5(w, I);  Homogenized([w], I);
  Homogenized(w, I);    -- don't just get the homogenizations of
                        -- the generators of I
Ideal(x^3 - yw^2, -xy + zw, x^2z - y^2w, y^3 - xz^2)
-------------------------------
  Homogenized(w,[[I,y-z^2],z-y^4]);
[[Ideal(x^3 - yw^2, -xy + zw, x^2z - y^2w, y^3 - xz^2), -z^2 + yw], -y^4 + zw^3]
-------------------------------

Syntax

Homogenized(X:INDET,E:T):T

where T is of type IDEAL or POLY, or T is a LIST recursively
constructed of types IDEAL, POLY, and LIST.

GBasis5, and more

   <type>groebner</type>
   <type>groebner-basic</type>
   <type>ideal</type>
   <type>polynomial</type>
   <type>cocoaserver</type>