## Cyclotomic points on curves.(English)Zbl 1029.11009

Bennett, M. A. (ed.) et al., Number theory for the millennium I. Proceedings of the millennial conference on number theory, Urbana-Champaign, IL, USA, May 21-26, 2000. Natick, MA: A K Peters. 67-85 (2002).
Let $$f(x,y)$$ be a Laurent polynomial with complex coefficients, and let $$V(f)$$ be the area of its Newton polytope. A pair $$(a,b)$$ is said to be a cyclotomic point if both $$a$$ and $$b$$ are roots of unity. Suppose that $$f$$ is such that there are only finitely many cyclotomic points $$(a,b)$$ for which $$f(a,b)=0.$$ The authors prove that then on the curve $$f(x,y)=0$$ there are at most $$22 V(f)$$ cyclotomic points. They give an example of $f(x,y)=xy+1/xy+x+1/x+y+1/y+1$ which shows that the constant $$22$$ cannot be replaced by a constant smaller than $$16.$$ An infinite family of polynomials for which the constant is approximately $$10$$ is also constructed. Finally, the authors give a sharp version of their upper bound for the number of cyclotomic points on a curve. Since one needs extra work to compute this sharp bound, the bound $$22 V(f)$$ seems more practical for applications. In the paper, the authors also give an algorithm for finding the cyclotomic part of a polynomial in one variable and the literature where one can find much more general (but not so sharp) results.
For the entire collection see [Zbl 1002.00005].

### MSC:

 11C08 Polynomials in number theory 11R09 Polynomials (irreducibility, etc.)

### Keywords:

roots of unity; polynomials; Newton polytope