Proof that two curves of odd degree have a common point in $\mathbb{R}P^2$.

Question

Please help me to prove that two curves of odd degree have a common point in $\mathbb{R}P^2$.

My thoughts:

This is a statement about real homogenous polynomials in three variables of odd degree. Unfortunately I do not know any theorem that can guarantee existence of a root of a multivariable polynomial. But there is a theorem about existent of a root of single variable polynomial of odd degree. Maybe I should somehow use that theorem?

Thanks a lot for your help.

Answer 1

Make your curves affine (by setting $z=1$). Now, compute the resultant of the two polynomials. This will have degree equal to the product of the degrees, which is odd, which means it will have a real root, which gives you a common point (if you don't know about resultants, google).