# If the roots of unity lie on a circle, do arbitrary polynomial roots also lie on some kind of characteristic curve?

I've been learning about roots of unity and how they manifest on the complex plane. I understand that if you take $$z^{n}=1$$, then the values of $$z$$ that satisfy this equation happen to lie on the unit circle with equal angles $$\frac{2\pi}{n}$$ between them.

I tried a couple examples on Wolfram Alpha, here's z^5 = 1 and z^12 = 1. But I was wondering if a similar strategy could be used for arbitrary complex polynomials, so I decided to tweak the inputs to have more terms. For example, here's z^5 + z^3 = 1 and z^5 - z^3 + z = 1.

The complex plane representations generated by Wolfram Alpha seem to suggest that the solutions might lie on an ellipse instead of a pure circle, or maybe some other characteristic curve. Is this the case?

• At least for a quintic, the points lie on a conic, although not necesseraly an elipse: it can be a circle, parabolla or even hyperbola. I do not know where there is some formula for higher degree polynomials, good question. Commented Apr 15, 2021 at 10:04
• Not in general. But see the nice pictures in math.stackexchange.com/questions/535720/…, especially math.stackexchange.com/a/109605/589
– lhf
Commented Apr 15, 2021 at 10:10
• @RicardoMM I see, interesting that it's known for quintics specifically but not necessarily for other polynomials. Can you recommend me any resources to learn more about complex analysis (granted I get more familiar with complex numbers first)? Commented Apr 15, 2021 at 10:12
• @JansthcirlU I think Conway's book is a nice textbook books.google.ca/books?id=9LtfZr1snG0C&hl=pt-BR. Of course it depends on your field of study. Commented Apr 15, 2021 at 10:30
• @JansthcirlU In special cases there may be a sort of symmetry, but I do not know under what conditions we have that. See the comment to my answer. Commented Apr 16, 2021 at 22:14