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Bobby Lee

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- 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.

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.