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.

  • 1
    $\begingroup$ I think that what you are looking for is Bezout's Theorem (en.wikipedia.org/wiki/Bézout%27s_theorem)? $\endgroup$ – An Coileanach Mar 9 '18 at 20:02
  • $\begingroup$ This is the approach I’d have used, since I understand Bézout and resultants are a mystery to me. $\endgroup$ – Lubin Mar 9 '18 at 21:52

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).

  • $\begingroup$ Thanks a lot! I will learn about resultants, but I already see that the solution is correct. $\endgroup$ – Hedgehog Mar 9 '18 at 20:14

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