make a polynomial in two variables homogeneous in three variables
homogeneous(f, x, y, z)
polynomial in x and y
x, y, z
Algebraic curves are often represented using a polynomial in two variables, or as a homogeneous polynomial in three variables. This command converts a polynomial to the homogeneous representation.
If f is a polynomial in x and y of degree d, and f can be written as ∑i,jci,j⁢xi⁢yj for some coefficients ci,j, then the output of this command is ∑i,jci,j⁢xi⁢yj⁢zd−i−j.
f ≔ y2−x3
Now one can convert back as follows:
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