# Questions about algebraic curves and homogeneous polynomial equations

• I
Bobby Lee
TL;DR Summary
Here, I present a few questions on the concept of homogeneous polynomial equations and their connection with algebraic curves. Since the latter is a topic of great interest, I think these questions as well as their answers might be of interest to those who subscribe to PhysicsForums.
It is generally well-known that a plane algebraic curve is a curve in ##\mathcal{CP}^{2}## given by a homogeneous polynomial equation ##f(x,y)= \sum^{N}_{i+j=0}a_{i\,j}x^{i}y^{j}=0##, where ##i## and ##j## are nonnegative integers and not all coefficients ##a_{ij}## are zero~[1].
In addition, if ##f(x,y)## is a homogeneous polynomial of degree ##d##, then [2]
$$f(tx,ty)=t^{d}f(x,y)$$
With those definitions before us, may we say that the following equations,
$$320 x^4-512 x^3 y+176 x^3+272 x^2 y^2-184 x^2 y+34 x^2$$
$$-96 x y^3+160 x y^2-84 x y+14 x+160 y^4-272 y^3+172 y^2-48 y+5=0$$
and
$$25 x^2-120 x y-10 x+244 y^2+124 y+26=0$$
are algebraic curves? if so, how may I prove it using the definitions presented above?

References

1. Jennings, G. A. (2012). Modern Geometry with Applications. Estados Unidos: Springer New York.
2. Prasolov, V. V., Lando, S. K., Kazaryan, M. E. (2019). Algebraic Curves: Towards Moduli Spaces. Alemanha: Springer International Publishing.

Staff Emeritus
Gold Member
What makes you worried that these do not immediately follow the definition you give for an algebraic curve? If it makes you feel better you can write out for the first one if meets the definition with ##a_{4,0}=320## etc, but no one would be this pedantic usually.

Bobby Lee
Gold Member
It is generally well-known that a plane algebraic curve is a curve in ##\mathcal{CP}^{2}## given by a homogeneous polynomial equation ##f(x,y)= \sum^{N}_{i+j=0}a_{i\,j}x^{i}y^{j}=0##, where ##i## and ##j## are nonnegative integers and not all coefficients ##a_{ij}## are zero~[1].
This defines a (finite) subset of ##\mathbb{C}P^1##, not ##\mathbb{C}P^2.## An algebraic curve in ##\mathbb{C}P^2## could be described as the zero set of a homogenous polynomial in ##3## variables.

When you have a possibly nonhomoenous polynomial in ##n## variables, you can "homogenize" it by multiplying each term by an appropriate power of a new variable. For example, the equation ##x^2+y+1=0## defines a curve in the affine plane, but ##x^2+yz+z^2=0## defines a curve in the projective plane. These curves are related because if you take the points ##[x:y:z]## where ##z\neq 0## and identify it with the affine plane by scaling ##z## to be ##1##, then the two curves coincide.

Bobby Lee and fresh_42